Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia

Chapter 5 Pairs of Random Varibales - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $X$ be the maximum and let $Y$ be the minimum of the number of heads obtained when Carlos and Michael each flip a fair coin twice.
(a) Describe the underlying space $S$ of this random experiment and show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the probabilities for all values of $(X, Y)$.
(c) Find $P[X=Y]$.
(d) Repeat parts b and c if Carlos uses a biased coin with $P[$ heads $]=3 / 4$.

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Problem 2

Let $X$ be the difference and let $Y$ be the sum of the number of heads obtained when Carlos and Michael each flip a fair coin twice.
(a) Describe the underlying space $S$ of this random experiment and show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the probabilities for all values of $(X, Y)$.
(c) Find $P[X+Y=1], P[X+Y=2]$.

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Problem 3

The input $X$ to a communication channel is " -1 "or " 1 ", with respective probabilities $1 / 4$ and $3 / 4$. The output of the channel $Y$ is equal to: the corresponding input $X$ with probability $1-p-p_e ;-X$ with probability $p ; 0$ with probability $p_e$.
(a) Describe the underlying space $S$ of this random experiment and show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the probabilities for all values of $(X, Y)$.
(c) Find $P[X \neq Y], P[Y=0]$.

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03:11

Problem 4

(a) Specify the range of the pair $\left(N_1, N_2\right)$ in Example 5.2 .
(b) Specify and sketch the event "more revenue comes from type 1 requests than type 2 requests."

Maxime Rossetti

Numerade Educator

Problem 5

(a) Specify the range of the pair $(Q, R)$ in Example 5.3.
(b) Specify and sketch the event "last packet is more than half full."

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01:22

Problem 6

Let the pair of random variables $H$ and $W$ be the height and weight in Example 5.1. The body mass index is a measure of body fat and is defined by BMI $=W / H^2$ where $W$ is in kilograms and $H$ is in meters. Determine and sketch on the plane the following events: $A=\{$ "obese," BMI $\geq 30\} ; B=\{$ "overweight," $25 \leq \mathrm{BMI}<30\}$; $C=\{$ "normal," $18.5 \leq \mathrm{BMI}<25\}$; and $D=\{$ "underweight," $\mathrm{BMI}<18.5\}$.

Sherrie Fenner

Numerade Educator

06:22

Problem 7

Let $(X, Y)$ be the two-dimensional noise signal in Example 5.4. Specify and sketch the events:
(a) "Maximum noise magnitude is greater than 5 ."
(b) "The noise power $X^2+Y^2$ is greater than 4."
(c) "The noise power $X^2+Y^2$ is greater than 4 and less than 9."

Tatiana Graham

Numerade Educator

03:17

Problem 8

For the pair of random variables $(X, Y)$ sketch the region of the plane corresponding to the following events. Identify which events are of product form.
(a) $\{X+Y>3\}$.
(b) $\left\{e^X>Y e^3\right\}$.
(c) $\{\min (X, Y)>0\} \cup\{\max \{X, Y)<0\}$.
(d) $\{|X-Y| \geq 1\}$.
(e) $\{|X / Y|>2\}$.
(f) $\{X / Y<2\}$.
(g) $\left\{X^3>Y\right\}$.
(h) $\{X Y<0\}$.
(i) $\{\max (|X|, Y)<3\}$.

Linda Winkler

Numerade Educator

Problem 9

(a) Find and sketch $p_{X, Y}(x, y)$ in Problem 5.1 when using a fair coin.
(b) Find $p_X(x)$ and $p_Y(y)$.
(c) Repeat parts a and b if Carlos uses a biased coin with $P[$ heads $]=3 / 4$.

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Problem 10

(a) Find and sketch $p_{X, Y}(x, y)$ in Problem 5.2 when using a fair coin.
(b) Find $p_X(x)$ and $p_Y(y)$.
(c) Repeat parts a and b if Carlos uses a biased coin with $P$ [heads] $=3 / 4$.

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02:21

Problem 11

(a) Find the marginal pmf's for the pairs of random variables with the indicated joint pmf.
Table can't copy
(b) Find the probability of the events $A=\{X>0\}, B=\{X \geq Y\}$, and $C=$ $\{X=-Y\}$ for the above joint pmf's.

Amany Waheeb

Numerade Educator

Problem 12

A modem transmits a two-dimensional signal $(X, Y)$ given by:

$$
X=r \cos (2 \pi \Theta / 8) \quad \text { and } \quad Y=r \sin (2 \pi \Theta / 8)
$$

where $\Theta$ is a discrete uniform random variable in the set $\{0,1,2, \ldots, 7\}$.
(a) Show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the joint pmf of $X$ and $Y$.
(c) Find the marginal pmf of $X$ and of $Y$.
(d) Find the probability of the following events: $A=\{X=0\}, B=\{Y \leq r / \sqrt{2}\}$, $C=\{X \geq r / \sqrt{2}, Y \geq r / \sqrt{2}\}, D=\{X<-r / \sqrt{2}\}$.

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04:04

Problem 13

Let $N_1$ be the number of Web page requests arriving at a server in a $100-\mathrm{ms}$ period and let $N_2$ be the number of Web page requests arriving at a server in the next $100-\mathrm{ms}$ period. Assume that in a $1-\mathrm{ms}$ interval either zero or one page request takes place with respective probabilities $1-p=0.95$ and $p=0.05$, and that the requests in different $1-\mathrm{ms}$ intervals are independent of each other.
(a) Describe the underlying space $S$ of this random experiment and show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the joint pmf of $X$ and $Y$.
(c) Find the marginal pmf for $X$ and for $Y$.
(d) Find the probability of the events $A=\{X \geq Y\}, B=\{X=Y=0\}, C=\{X>5$. $Y>3\}$.
(e) Find the probability of the event $D=\{X+Y=10\}$.

James Kiss

Numerade Educator

12:15

Problem 14

Let $N_1$ be the number of Web page requests arriving at a server in the period $(0,100) \mathrm{ms}$ and let $N_2$ be the total combined number of Web page requests arriving at a server in the period $(0,200) \mathrm{ms}$. Assume arrivals occur as in Problem 5.13.
(a) Describe the underlying space $S$ of this random experiment and show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the joint pmf of $N_1$ and $N_2$.
(c) Find the marginal pmf for $N_1$ and $N_2$.
(d) Find the probability of the events $A=\left\{N_1<N_2\right\}, B=\left\{N_2=0\right\}, C=\left\{N_1>5\right.$, $\left.N_2>3\right\}, D=\left\{\left|N_2-2 N_1\right|<2\right\}$.

Robin Corrigan

Numerade Educator

Problem 15

At even time instants, a robot moves either $+\Delta \mathrm{cm}$ or $-\Delta \mathrm{cm}$ in the $x$-direction according to the outcome of a coin flip; at odd time instants, a robot moves similarly according to another coin flip in the $y$-direction. Assuming that the robot begins at the origin, let $X$ and $Y$ be the coordinates of the location of the robot after $2 n$ time instants.
(a) Describe the underlying space $S$ of this random experiment and show the mapping from $S$ to $S_{X Y}$, the range of the pair $(X, Y)$.
(b) Find the marginal pmf of the coordinates $X$ and $Y$.
(c) Find the probability that the robot is within distance $\sqrt{2}$ of the origin after $2 n$ time instants.

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Problem 16

(a) Sketch the joint cdf for the pair $(X, Y)$ in Problem 5.1 and verify that the properties of the joint cdf are satisfied. You may find it helpful to first divide the plane into regions where the cdf is constant.
(b) Find the marginal cdf of $X$ and of $Y$.

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00:41

Problem 17

A point $(X, Y)$ is selected at random inside a triangle defined by $\{(x, y): 0 \leq y \leq x \leq 1\}$. Assume the point is equally likely to fall anywhere in the triangle.
(a) Find the joint cdf of $X$ and $Y$.
(b) Find the marginal cdf of $X$ and of $Y$.
(c) Find the probabilities of the following events in terms of the joint odf: $A=\{X \leq 1 / 2, Y \leq 3 / 4\} ; B=\{1 / 4<X \leq 3 / 4,1 / 4<Y \leq 3 / 4\}$.

Victor Salazar

Numerade Educator

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Problem 18

A dart is equally likely to land at any point ( $X_1, X_2$ ) inside a circular target of unit radius Let $R$ and $\Theta$ be the radius and angle of the point ( $X_1, X_2$ ).
(a) Find the joint $\operatorname{cdf}$ of $R$ and $\Theta$.
(b) Find the marginal cdf of $R$ and $\Theta$.
(c) Use the joint cdf to find the probability that the point is in the first quadrant of the real plane and that the radius is greater than 0.5 .

Victor Salazar

Numerade Educator

Problem 19

Find an expression for the probability of the events in Problem 5.8 parts $\mathrm{c}, \mathrm{h}$, and i in terms of the joint cdf of $X$ and $Y$.

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01:05

Problem 20

The pair $(X, Y)$ has joint cdf given by:

$$
F_{X, Y}(x, y)= \begin{cases}\left(1-1 / x^2\right)\left(1-1 / y^2\right) & \text { for } x>1, y>1 \\ 0 & \text { elsewhere. }\end{cases}
$$

(a) Sketch the joint cdf.
(b) Find the marginal cdf of $X$ and of $Y$.
(c) Find the probability of the following events: $\{X<3, Y \leq 5\},\{X>4, Y>3\}$.

Hast Aggarwal

Numerade Educator

05:13

Problem 21

Is the following a valid cdf? Why?

$$
F_{X, Y}(x, y)= \begin{cases}\left(1-1 / x^2 y^2\right) & \text { for } x>1, y>1 \\ 0 & \text { elsewhere. }\end{cases}
$$

Uma Kumari

Numerade Educator

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Problem 22

Let $F_X(x)$ and $F_Y(y)$ be valid one-dimensional cdf's. Show that $F_{X, Y}(x, y)=F_X(x) F_Y(y)$ satisfies the properties of a two-dimensional cdf.

Victor Salazar

Numerade Educator

Problem 23

The number of users logged onto a system $N$ and the time $T$ until the next user logs off have joint probability given by:

$$
P[N=n, X \leq t]=(1-\rho) \rho^{n-1}\left(1-e^{-n \lambda t}\right) \quad \text { for } n=1,2, \ldots \quad t>0 \text {. }
$$

(a) Sketch the above joint probability.
(b) Find the marginal pmf of $N$.
(c) Find the marginal cdf of $X$.
(d) Find $P[N \leq 3, X>3 / \lambda]$.

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01:28

Problem 24

A factory has $n$ machines of a certain type. Let $p$ be the probability that a machine is working on any given day, and let $N$ be the total number of machines working on a certain day. The time $T$ required to manufacture an item is an exponentially distributed random variable with rate $k \alpha$ if $k$ machines are working. Find and $P[T \leq t]$. Find $P[T \leq t]$ as $t \rightarrow \infty$ and explain the result.

Dushyant Barot

Numerade Educator

11:06

Problem 25

The amplitudes of two signals $X$ and $Y$ have joint pdf:

$$
f_{X, Y}(x, y)=e^{-x / 2} y e^{-y^2} \quad \text { for } x>0, y>0 \text {. }
$$

(a) Find the joint cdf.
(b) Find $P\left[X^{1 / 2}>Y\right]$.
(c) Find the marginal pdfs.

Sam Low

Numerade Educator

00:59

Problem 26

Let $X$ and $Y$ have joint pdf:

$$
f_{X, Y}(x, y)=k(x+y) \quad \text { for } 0 \leq x \leq 1,0 \leq y \leq 1 .
$$

(a) Find $k$.
(b) Find the joint cdf of $(X, Y)$.
(c) Find the marginal pdf of $X$ and of $Y$.
(d) Find $P[X<Y], P\left[Y<X^2\right], P[X+Y>0.5]$.

Victor Salazar

Numerade Educator

00:59

Problem 27

Let $X$ and $Y$ have joint pdf:

$$
f_{X, Y}(x, y)=k x(1-x) y \quad \text { for } 0<x<1,0<y<1
$$

(a) Find $k$.
(b) Find the joint cdf of $(X, Y)$.
(c) Find the marginal pdf of $X$ and of $Y$.
(d) Find $P\left[Y<X^{1 / 2}\right], P[X<Y]$.

Victor Salazar

Numerade Educator

01:05

Problem 28

The random vector $(X, Y)$ is uniformly distributed (i.e., $f(x, y)=k$ ) in the regions shown in Fig. P5.1 and zero elsewhere.
FIGURE P5.1 can't copy
(a) Find the value of $k$ in each case.
(b) Find the marginal pdf for $X$ and for $Y$ in each case.
(c) Find $P[X>0, Y>0]$.

Hast Aggarwal

Numerade Educator

23:27

Problem 29

(a) Find the joint cdf for the vector random variable introduced in Example 5.16.
(b) Use the result of part a to find the marginal cdf of $X$ and of $Y$.

Chris Trentman

Numerade Educator

09:12

Problem 30

Let $X$ and $Y$ have the joint pdf:

$$
f_{X, Y}(x, y)=y e^{-y(1+x)} \quad \text { for } x>0, y>0
$$

Find the marginal pdf of $X$ and of $Y$.

Abhirup Pal

Numerade Educator

01:05

Problem 31

Let $X$ and $Y$ be the pair of random variables in Problem 5.17.
(a) Find the joint pdf of $X$ and $Y$.
(b) Find the marginal pdf of $X$ and of $Y$.
(c) Find $P\left[Y<X^2\right]$.

Hast Aggarwal

Numerade Educator

Problem 32

Let $R$ and $\Theta$ be the pair of random variables in Problem 5.18.
(a) Find the joint pdf of $R$ and $\Theta$.
(b) Find the marginal pdf of $R$ and of $\Theta$.

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03:35

Problem 33

Let $(X, Y)$ be the jointly Gaussian random variables discussed in Example 5.18. Find $P\left[X^2+Y^2>r^2\right]$ when $\rho=0$. Hint: Use polar coordinates to compute the integral.

Prabhakar Kumar

Numerade Educator

Problem 34

The general form of the joint pdf for two jointly Gaussian random variables is given by Eq. (5.61a). Show that $X$ and $Y$ have marginal pdfs that correspond to Gaussian random variables with means $m_1$ and $m_2$ and variances $\sigma_1^2$ and $\sigma_2^2$ respectively.

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Problem 35

The input $X$ to a communication channel is +1 or -1 with probability $p$ and $1-p$, respectively. The received signal $Y$ is the sum of $X$ and noise $N$ which has a Gaussian distribution with zero mean and variance $\sigma^2=0.25$.
(a) Find the joint probability $P[X=j, Y \leq y]$.
(b) Find the marginal pmf of $X$ and the marginal pdf of $Y$.
(c) Suppose we are given that $Y>0$. Which is more likely, $X=1$ or $X=-1$ ?

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Problem 36

A modem sends a two-dimensional signal $\mathbf{X}$ from the set $\{(1,1),(1,-1),(-1,1)$, $(-1,-1)\}$. The channel adds a noise signal $\left(N_1, N_2\right)$, so the received signal is $\mathbf{Y}=\mathbf{X}+\mathbf{N}=\left(X_1+N_1, X_2+N_2\right)$. Assume that $\left(N_1, N_2\right)$ have the jointly Gaussian pdf in Example 5.18 with $\rho=0$. Let the distance between $\mathbf{X}$ and $\mathbf{Y}$ be $d(\mathbf{X}, \mathbf{Y})=\left\{\left(X_1-Y_1\right)^2+\left(X_2-Y_2\right)^2\right\}^{1 / 2}$.
(a) Suppose that $\mathbf{X}=(1,1)$. Find and sketch region for the event $\{\mathbf{Y}$ is closer to $(1,1)$ than to the other possible values of $\mathbf{X}\}$. Evaluate the probability of this event.
(b) Suppose that $\mathbf{X}=(1,1)$. Find and sketch region for the event $\{\mathbf{Y}$ is closer to $(1,-1)$ than to the other possible values of $\mathbf{X}\}$. Evaluate the probability of this event.
(c) Suppose that $\mathbf{X}=(1,1)$. Find and sketch region for the event $\{d(\mathbf{X}, \mathbf{Y})>1\}$. Evaluate the probability of this event. Explain why this probability is an upper bound on the probability that $\mathbf{Y}$ is closer to a signal other than $\mathbf{X}=(1,1)$.

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06:33

Problem 37

Let $X$ be the number of full pairs and let $Y$ be the remainder of the number of dots observed in a toss of a fair die. Are $X$ and $Y$ independent random variables?

Amany Waheeb

Numerade Educator

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Problem 38

Let $X$ and $Y$ be the coordinates of the robot in Problem 5.15 after $2 n$ time instants. Determine whether $X$ and $Y$ are independent random variables.

Victor Salazar

Numerade Educator

04:00

Problem 39

Let $X$ and $Y$ be the coordinates of the two-dimensional modem signal $(X, Y)$ in Problem 5.12.
(a) Determine if $X$ and $Y$ are independent random variables.
(b) Repeat part a if even values of $\Theta$ are twice as likely as odd values.

Amany Waheeb

Numerade Educator

01:33

Problem 40

Determine which of the joint pmfs in Problem 5.11 correspond to independent pairs of random variables.

Manik Pulyani

Numerade Educator

01:52

Problem 41

Michael takes the $7: 30$ bus every morning. The arrival time of the bus at the stop is uniformly distributed in the interval $[7: 27,7: 37]$. Michael's arrival time at the stop is also uniformly distributed in the interval [7:25,7:40]. Assume that Michael's and the bus's arrival times are independent random variables.
(a) What is the probability that Michael arrives more than 5 minutes before the bus?
(b) What is the probability that Michael misses the bus?

Amany Waheeb

Numerade Educator

01:39

Problem 42

Are $R$ and $\Theta$ independent in Problem 5.18?

Michelle Z.

Numerade Educator

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Problem 43

Are $X$ and $Y$ independent in Problem 5.20?

Victor Salazar

Numerade Educator

Problem 44

Are the signal amplitudes $X$ and $Y$ independent in Problem 5.25?

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Problem 45

Are $X$ and $Y$ independent in Problem 5.26?

Victor Salazar

Numerade Educator

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Problem 46

Are $X$ and $Y$ independent in Problem 5.27?

Victor Salazar

Numerade Educator

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Problem 47

Let $X$ and $Y$ be independent random variables. Find an expression for the probability of the following events in terms of $F_X(x)$ and $F_Y(y)$.
(a) $\{a<X \leq b\} \cap\{Y>d\}$.
(b) $\{a<X \leq b\} \cap\{c \leq Y<d\}$.
(c) $\{|X|<a\} \cap\{c \leq Y \leq d\}$.

Victor Salazar

Numerade Educator

Problem 48

Let $X$ and $Y$ be independent random variables that are uniformly distributed in $[-1,1]$. Find the probability of the following events:
(a) $P\left[X^2<1 / 2,|Y|<1 / 2\right]$.
(b) $P[4 X<1, Y<0]$.
(c) $P[X Y<1 / 2]$.
(d) $P[\max (X, Y)<1 / 3]$.

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08:59

Problem 49

Let $X$ and $Y$ be random variables that take on values from the set $\{-1,0,1\}$.
(a) Find a joint pmf for which $X$ and $Y$ are independent.
(b) Are $X^2$ and $Y^2$ independent random variables for the pmf in part a?
(c) Find a joint pmf for which $X$ and $Y$ are not independent, but for which $X^2$ and $Y^2$ are independent.

Ekaveera Kumar

Numerade Educator

04:06

Problem 50

Let $X$ and $Y$ be the jointly Gaussian random variables introduced in Problem 5.34.
(a) Show that $X$ and $Y$ are independent random variables if and only if $\rho=0$.
(b) Suppose $\rho=0$, find $P[X Y<0]$.

Amany Waheeb

Numerade Educator

01:45

Problem 51

Two fair dice are tossed repeatedly until a pair occurs. Let $K$ be the number of tosses required and let $X$ be the number showing up in the pair. Find the joint pmf of $K$ and $X$ and determine whether $K$ and $X$ are independent.

Manik Pulyani

Numerade Educator

Problem 52

The number of devices $L$ produced in a day is geometric distributed with probability of success $p$. Let $N$ be the number of working devices and let $M$ be the number of defective devices produced in a day.
(a) Are $N$ and $M$ independent random variables?
(b) Find the joint pmf of $N$ and $M$.
(c) Find the marginal pmfs of $N$ and $M$. (See hint in Problem 5.87b.)
(d) Are $L$ and $M$ independent random variables?

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Problem 53

Let $N_1$ be the number of Web page requests arriving at a server in a $100-\mathrm{ms}$ period and let $N_2$ be the number of Web page requests arriving at a server in the next $100-\mathrm{ms}$ period. Use the result of Problem 5.13 parts a and b to develop a model where $N_1$ and $N_2$ are independent Poisson random variables.

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Problem 54

(a) Show that Eq. (5.22) implies Eq. (5.21).
(b) Show that Eq. (5.21) implies Eq. (5.22).

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Problem 55

Verify that Eqs. (5.22) and (5.23) can be obtained from each other.

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04:05

Problem 56

(a) Find $E\left[(X+Y)^2\right]$.
(b) Find the variance of $X+Y$.
(c) Under what condition is the variance of the sum equal to the sum of the individual variances?

Charles Machakwa

Numerade Educator

07:17

Problem 57

Find $E[|X-Y|]$ if $X$ and $Y$ are independent exponential random variables with parameters $\lambda_1=1$ and $\lambda_2=2$, respectively.

Ekaveera Kumar

Numerade Educator

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Problem 58

Find $E\left[X^2 e^Y\right]$ where $X$ and $Y$ are independent random variables, $X$ is a zero-mean, unit-variance Gaussian random variable, and $Y$ is a uniform random variable in the interval $[0,3]$.

Victor Salazar

Numerade Educator

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Problem 59

For the discrete random variables $X$ and $Y$ in Problem 5.1, find the correlation and covariance, and indicate whether the random variables are independent, orthogonal, or uncorrelated.

Victor Salazar

Numerade Educator

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Problem 60

For the discrete random variables $X$ and $Y$ in Problem 5.2, find the correlation and covariance, and indicate whether the random variables are independent, orthogonal, or uncorrelated.

Victor Salazar

Numerade Educator

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Problem 61

For the three pairs of discrete random variables in Problem 5.11, find the correlation and covariance of $X$ and $Y$, and indicate whether the random variables are independent, orthogonal, or uncorrelated.

Victor Salazar

Numerade Educator

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Problem 62

Let $N_1$ and $N_2$ be the number of Web page requests in Problem 5.13. Find the correlation and covariance of $N_1$ and $N_2$, and indicate whether the random variables are independent, orthogonal, or uncorrelated.

Victor Salazar

Numerade Educator

Problem 63

Repeat Problem 5.62 for $N_1$ and $N_2$, the number of Web page requests in Problem 5.14.

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Problem 64

Let $N$ and $T$ be the number of users logged on and the time till the next logoff in Problem 5.23. Find the correlation and covariance of $N$ and $T$, and indicate whether the random variables are independent, orthogonal, or uncorrelated.

Victor Salazar

Numerade Educator

Problem 65

Find the correlation and covariance of $X$ and $Y$ in Problem 5.26. Determine whether $X$ and $Y$ are independent, orthogonal, or uncorrelated.

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03:49

Problem 66

Repeat Problem 5.65 for $X$ and $Y$ in Problem 5.27.

Ma. Theresa Alin

Numerade Educator

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Problem 67

For the three pairs of continuous random variables $X$ and $Y$ in Problem 5.29, find the correlation and covariance, and indicate whether the random variables are independent, orthogonal, or uncorrelated.

Victor Salazar

Numerade Educator

04:33

Problem 68

Find the correlation coefficient between $X$ and $Y=a X+b$. Does the answer depend on the sign of $a$ ?

Chris Trentman

Numerade Educator

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Problem 69

Propose a method for estimating the covariance of two random variables.

Eduard Sanchez

Numerade Educator

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Problem 70

(a) Complete the calculations for the correlation coefficient in Example 5.27.
(b) Repeat the calculations if $X$ and $Y$ have the pdf:

$$
f_{X, Y}(x, y)=e^{-(x+|y|)} \quad \text { for } x>0,-x<y<x .
$$

Victor Salazar

Numerade Educator

Problem 71

The output of a channel $Y=X+N$, where the input $X$ and the noise $N$ are independent, zero-mean random variables.
(a) Find the correlation coefficient between the input $X$ and the output $Y$.
(b) Suppose we estimate the input $X$ by a linear function $g(Y)=a Y$. Find the value of $a$ that minimizes the mean squared error $E\left[(X-a Y)^2\right]$.
(c) Express the resulting mean-square error in terms of $\sigma_X / \sigma_N$.

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Problem 72

In Example 5.27 let $X=\cos \Theta / 4$ and $Y=\sin \Theta / 4$. Are $X$ and $Y$ uncorrelated?

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01:03

Problem 73

(a) Show that $\operatorname{COV}(X, E[Y \mid X])=\operatorname{COV}(X, Y)$.
(b) Show that $E[Y \mid X=x]=E[Y]$, for all $x$, implies that $X$ and $Y$ are uncorrelated.

Amany Waheeb

Numerade Educator

03:38

Problem 74

Use the fact that $E\left[(t X+Y)^2\right] \geq 0$ for all $t$ to prove the Cauchy-Schwarz inequality:

$$
(E[X Y])^2 \leq E\left[X^2\right] E\left[Y^2\right]
$$

Hint: Consider the discriminant of the quadratic equation in $t$ that results from the above inequality.

Amany Waheeb

Numerade Educator

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Problem 75

(a) Find $p_Y(y \mid x)$ and $p_X(x \mid y)$ in Problem 5.1 assuming fair coins are used.
(b) Find $p_Y(y \mid x)$ and $p_X(x \mid y)$ in Problem 5.1 assuming Carlos uses a coin with $p=3 / 4$.
(c) What is the effect on $p_X(x \mid y)$ of Carlos using a biased coin?
(d) Find $E[Y \mid X=x]$ and $E[X \mid Y=y]$ in part a; then find $E[X]$ and $E[Y]$.
(e) Find $E[Y \mid X=x]$ and $E[X \mid Y=y]$ in part b; then find $E[X]$ and $E[Y]$.

Rashmi Sinha

Numerade Educator

Problem 76

(a) Find $p_X(x \mid y)$ for the communication channel in Problem 5.3.
(b) For each value of $y$, find the value of $x$ that maximizes $p_X(x \mid y)$. State any assumptions about $p$ and $p_e$.
(c) Find the probability of error if a receiver uses the decision rule from part b.

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Problem 77

(a) In Problem 5.11(i), which conditional pmf given $X$ provides the most information about $Y: p_Y(y \mid-1), p_Y(y \mid 0)$, or $p_Y(y \mid+1)$ ? Explain why.
(b) Compare the conditional pmfs in Problems 5.11(ii) and (iii) and explain which of these two cases is "more random."
(c) Find $E[Y \mid X=x]$ and $E[X \mid Y=y]$ in Problems 5.11(i), (ii), (iii); then find $E[X]$ and $E[Y]$.
(d) Find $E\left[Y^2 \mid X=x\right]$ and $E\left[X^2 \mid Y=y\right]$ in Problems 5.11(i), (ii), (iii); then find $\operatorname{VAR}[X]$ and $\operatorname{VAR}[Y]$.

Rashmi Sinha

Numerade Educator

02:08

Problem 78

(a) Find the conditional pmf of $N_1$ given $N_2$ in Problem 5.14 .
(b) Find $P\left[N_1=k \mid N_2=2 k\right]$ for $k=5,10,20$. Hint: Use Stirling's fromula.
(c) Find $E\left[N_1 \mid N_2=k\right]$, then find $E\left[N_1\right]$.

Hunza Gilgit

Numerade Educator

01:53

Problem 79

In Example 5.30, let $Y$ be the number of defects inside the region $R$ and let $Z$ be the number of defects outside the region.
(a) Find the pmf of $Z$ given $Y$.
(b) Find the joint pmf of $Y$ and $Z$.
(c) Are $Y$ and $Z$ independent random variables? Is the result intuitive?

Hast Aggarwal

Numerade Educator

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Problem 80

(a) Find $f_Y(y \mid x)$ in Problem 5.26 .
(b) Find $P[Y>X \mid x]$.
(c) Find $P[Y>X]$ using part b.
(d) Find $E[Y \mid X=x]$.

Rashmi Sinha

Numerade Educator

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Problem 81

(a) Find $f_Y(y \mid x)$ in Problem 5.28(i).
(b) Find $E[Y \mid X=x]$ and $E[Y]$.
(c) Repeat parts a and b of Problem 5.28(ii).
(d) Repeat parts a and b of Problem 5.28(iii).

Rashmi Sinha

Numerade Educator

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Problem 82

(a) Find $f_Y(y \mid x)$ in Example 5.27.
(b) Find $E[Y \mid X=x]$.
(c) Find $E[Y]$.
(d) Find $E[X Y \mid X=x]$.
(e) Find $E[X Y]$.

Rashmi Sinha

Numerade Educator

Problem 83

Find $f_Y(y \mid x)$ and $f_X(x \mid y)$ for the jointly Gaussian pdf in Problem 5.34.

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Problem 84

(a) Find $f_X(t \mid N=n)$ in Problem 5.23.
(b) Find $E\left[X^t \mid N=n\right]$.
(c) Find the value of $n$ that maximizes $P[N=n \mid t<X<t+d t]$.

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Problem 85

(a) Find $p_Y(y \mid x)$ and $p_X(x \mid y)$ in Problem 5.12.
(b) Find $E[Y \mid X=x]$.
(c) Find $E[X Y \mid X=x]$ and $E[Y X]$.

Rashmi Sinha

Numerade Educator

04:25

Problem 86

A customer enters a store and is equally likely to be served by one of three clerks. The time taken by clerk 1 is a constant random variable with mean two minutes; the time for clerk 2 is exponentially distributed with mean two minutes; and the time for clerk 3 is Pareto distributed with mean two minutes and $\alpha=2.5$.
(a) Find the pdf of $T$, the time taken to service a customer.
(b) Find $E[T]$ and VAR $[T]$.

Ahmad Reda

Numerade Educator

Problem 87

A message requires $N$ time units to be transmitted, where $N$ is a geometric random variable with pmf $p_i=(1-a) a^{i-1}, i=1,2, \ldots$. A single new message arrives during a time unit with probability $p$, and no messages arrive with probability $1-p$. Let $K$ be the number of new messages that arrive during the transmission of a single message.
(a) Find $E[K]$ and VAR $[K]$ using conditional expectation.
(b) Find the pmf of $K$. Hint: $(1-\beta)^{-(k+1)}=\sum_{n=k}^{\infty}\binom{n}{k} \beta^{n-k}$.
(c) Find the conditional pmf of $N$ given $K=k$.
(d) Find the value of $n$ that maximizes $P[N=n \mid X=k]$.

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Problem 88

The number of defects in a VLSI chip is a Poisson random variable with rate $r$. However, $r$ is itself a gamma random variable with parameters $\alpha$ and $\lambda$.
(a) Use conditional expectation to find $E[N]$ and $\operatorname{VAR}[N]$.
(b) Find the pmf for $N$, the number of defects.

Victor Salazar

Numerade Educator

10:39

Problem 89

(a) In Problem 5.35 , find the conditional pmf of the input $X$ of the communication channel given that the output is in the interval $y<Y \leq y+d y$.
(b) Find the value of $X$ that is more probable given $y<Y \leq y+d y$.
(c) Find an expression for the probability of error if we use the result of part b to decide what the input to the channel was.

Chris Trentman

Numerade Educator

07:16

Problem 90

Two toys are started at the same time each with a different battery. The first battery has a lifetime that is exponentially distributed with mean 100 minutes; the second battery has a Rayleigh-distributed lifetime with mean 100 minutes.
(a) Find the cdf to the time $T$ until the battery in a toy first runs out.
(b) Suppose that both toys are still operating after 100 minutes. Find the cdf of the time $T_2$ that subsequently elapses until the battery in a toy first runs out.
(c) In part b, find the cdf of the total time that elapses until a battery first fails.

Robin Corrigan

Numerade Educator

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Problem 91

(a) Find the cdf of the time that elapses until both batteries run out in Problem 5.90a.
(b) Find the cdf of the remaining time until both batteries run out in Problem 5.90b.

Rashmi Sinha

Numerade Educator

14:50

Problem 92

Let $K$ and $N$ be independent random variables with nonnegative integer values.
(a) Find an expression for the pmf of $M=K+N$.
(b) Find the pmf of $M$ if $K$ and $N$ are binomial random variables with parameters $(k, p)$ and $(n, p)$.
(c) Find the pmf of $M$ if $K$ and $N$ are Poisson random variables with parameters $\alpha_1$ and $\alpha_2$, respectively.

Mengchun Cai

Numerade Educator

Problem 93

The number $X$ of goals the Bulldogs score against the Flames has a geometric distribution with mean 2; the number of goals $Y$ that the Flames score against the Bulldogs is also geometrically distributed but with mean 4.
(a) Find the pmf of the $Z=X-Y$. Assume $X$ and $Y$ are independent.
(b) What is the probability that the Bulldogs beat the Flames? Tie the Flames?
(c) Find $E[Z]$.

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Problem 94

Passengers arrive at an airport taxi stand every minute according to a Bernoulli random variable. A taxi will not leave until it has two passengers.
(a) Find the pmf until the time $T$ when the taxi has two passengers.
(b) Find the pmf for the time that the first customer waits.

Victor Salazar

Numerade Educator

01:17

Problem 95

Let $X$ and $Y$ be independent random variables that are uniformly distributed in the interval $[0,1]$. Find the pdf of $Z=X Y$.

Hoan Nguyen

Numerade Educator

01:34

Problem 96

Let $X_1, X_2$, and $X_3$ be independent and uniformly distributed in $[-1,1]$.
(a) Find the cdf and pdf of $Y=X_1+X_2$.
(b) Find the cdf of $Z=Y+X_3$.

Manik Pulyani

Numerade Educator

05:39

Problem 97

Let $X$ and $Y$ be independent random variables with gamma distributions and parameters $\left(\alpha_1, \lambda\right)$ and $\left(\alpha_2, \lambda\right)$, respectively. Show that $Z=X+Y$ is gamma-distributed with parameters $\left(\alpha_1+\alpha_2, \lambda\right)$. Hint: See Eq. (4.59).

Heena Haldankar

Numerade Educator

03:31

Problem 98

Signals $X$ and $Y$ are independent. $X$ is exponentially distributed with mean 1 and $Y$ is exponentially distributed with mean 1 .
(a) Find the cdf of $Z=|X-Y|$.
(b) Use the result of part a to find $E[Z]$.

Amany Waheeb

Numerade Educator

08:14

Problem 99

The random variables $X$ and $Y$ have the joint pdf

$$
f_{X, Y}(x, y)=e^{-(x+y)} \quad \text { for } 0<y<x<1
$$

Find the pdf of $Z=X+Y$.

Mengchun Cai

Numerade Educator

06:50

Problem 100

Let $X$ and $Y$ be independent Rayleigh random variables with parameters $\alpha=\beta=1$. Find the pdf of $Z=X / Y$.

Aman Gupta

Numerade Educator

05:50

Problem 101

Let $X$ and $Y$ be independent Gaussian random variables that are zero mean and unit variance. Show that $Z=X / Y$ is a Cauchy random variable.

Tatiana Graham

Numerade Educator

Problem 102

Find the joint cdf of $W=\min (X, Y)$ and $Z=\max (X, Y)$ if $X$ and $Y$ are independent and $X$ is uniformly distributed in $[0,1]$ and $Y$ is uniformly distributed in $[0,1]$.

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03:34

Problem 103

Find the joint cdf of $W=\min (X, Y)$ and $Z=\max (X, Y)$ if $X$ and $Y$ are independent exponential random variables with the same mean.

Amany Waheeb

Numerade Educator

03:34

Problem 104

Find the joint cdf of $W=\min (X, Y)$ and $Z=\max (X, Y)$ if $X$ and $Y$ are the independent Pareto random variables with the same distribution.

Amany Waheeb

Numerade Educator

04:28

Problem 105

Let $W=X+Y$ and $Z=X-Y$.
(a) Find an expression for the joint pdf of $W$ and $Z$.
(b) Find $f_{W, Z}(z, w)$ if $X$ and $Y$ are independent exponential random variables with parameter $\lambda=1$.
(c) Find $f_{W, Z}(z, w)$ if $X$ and $Y$ are independent Pareto random variables with the same distribution.

Narayan Hari

Numerade Educator

03:46

Problem 106

The pair $(X, Y)$ is uniformly distributed in a ring centered about the origin and inner and outer radii $r_1<r_2$. Let $R$ and $\Theta$ be the radius and angle corresponding to $(X, Y)$. Find the joint pdf of $R$ and $\Theta$.

Mahnoor Khan

Numerade Educator

Problem 107

Let $X$ and $Y$ be independent, zero-mean, unit-variance Gaussian random variables. Let $V=a X+b Y$ and $W=c X+e Y$.
(a) Find the joint pdf of $V$ and $W$, assuming the transformation matrix $A$ is invertible.
(b) Suppose $A$ is not invertible. What is the joint pdf of $V$ and $W$ ?

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01:50

Problem 108

Let $X$ and $Y$ be independent Gaussian random variables that are zero mean and unit variance. Let $W=X^2+Y^2$ and let $\Theta=\tan ^{-1}(Y / X)$. Find the joint pdf of $W$ and $\Theta$.

Manik Pulyani

Numerade Educator

Problem 109

Let $X$ and $Y$ be the random variables introduced in Example 5.4. Let $R=\left(X^2+Y^2\right)^{1 / 2}$ and let $\Theta=\tan ^{-1}(Y / X)$.
(a) Find the joint pdf of $R$ and $\Theta$.
(b) What is the joint pdf of $X$ and $Y$ ?

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06:28

Problem 110

Let $X$ and $Y$ be jointly Gaussian random variables with pdf

$$
f_{X, Y}(x, y)=\frac{\exp \left\{-2 x^2-y^2 / 2\right\}}{2 \pi c} \text { for all } x, y
$$

Find $\operatorname{VAR}[X], \operatorname{VAR}[Y]$, and $\operatorname{COV}(X, Y)$.

Mengchun Cai

Numerade Educator

01:07

Problem 111

Let $X$ and $Y$ be jointly Gaussian random variables with pdf

$$
f_{X, Y}(x, y)=\frac{\exp \left\{\frac{-1}{2}\left[x^2+4 y^2-3 x y+3 y-2 x+1\right]\right\}}{2 \pi} \text { for all } x, y .
$$

Find $E[X], E[Y], \operatorname{VAR}[X], \operatorname{VAR}[Y]$, and $\operatorname{COV}(X, Y)$.

Amany Waheeb

Numerade Educator

01:50

Problem 112

Let $X$ and $Y$ be jointly Gaussian random variables with $E[Y]=0, \sigma_1=1, \sigma_2=2$, and $E[X \mid Y]=Y / 4+1$. Find the joint pdf of $X$ and $Y$.

Manik Pulyani

Numerade Educator

11:15

Problem 113

Let $X$ and $Y$ be zero-mean, independent Gaussian random variables with $\sigma^2=1$.
(a) Find the value of $r$ for which the probability that $(X, Y)$ falls inside a circle of radius $r$ is $1 / 2$.
(b) Find the conditional pdf of $(X, Y)$ given that $(X, Y)$ is not inside a ring with inner radius $r_1$ and outer radius $r_2$.

Chris Trentman

Numerade Educator

Problem 114

Use a plotting program (as provided by Octave or MATLAB) to show the pdf for jointly Gaussian zero-mean random variables with the following parameters:
(a) $\sigma_1=1, \sigma_2=1, \rho=0$.
(b) $\sigma_1=1, \sigma_2=1, \rho=0.8$.
(c) $\sigma_1=1, \sigma_2=1, \rho=-0.8$.
(d) $\sigma_1=1, \sigma_2=2, \rho=0$.
(e) $\sigma_1=1, \sigma_2=2, \rho=0.8$.
(f) $\sigma_1=1, \sigma_2=10, \rho=0.8$.

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Problem 115

Let $X$ and $Y$ be zero-mean, jointly Gaussian random variables with $\sigma_1=1, \sigma_2=2$, and correlation coefficient $\rho$.
(a) Plot the principal axes of the constant-pdf ellipse of $(X, Y)$.
(b) Plot the conditional expectation of $Y$ given $X=x$.
(c) Are the plots in parts a and b the same or different? Why?

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Problem 116

Let $X$ and $Y$ be zero-mean, unit-variance jointly Gaussian random variables for which $\rho=1$. Sketch the joint cdf of $X$ and $Y$. Does a joint pdf exist?

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Problem 117

Let $h(x, y)$ be a joint Gaussian pdf for zero-mean, unit-variance Gaussian random variables with correlation coefficient $\rho_1$. Let $g(x, y)$ be a joint Gaussian pdf for zero-mean, unit-variance Gaussian random variables with correlation coefficient $\rho_2 \neq \rho_1$. Suppose the random variables $X$ and $Y$ have joint pdf

$$
f_{X, Y}(x, y)=\{h(x, y)+g(x, y)\} / 2
$$

(a) Find the marginal pdf for $X$ and for $Y$.
(b) Explain why $X$ and $Y$ are not jointly Gaussian random variables.

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07:53

Problem 118

Use conditional expectation to show that for $X$ and $Y$ zero-mean, jointly Gaussian random variables, $E\left[X^2 Y^2\right]=E\left[X^2\right] E\left[Y^2\right]+2 E[X Y]^2$.

Bryan Lynn

Numerade Educator

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Problem 119

Let $\mathbf{X}=(X, Y)$ be the zero-mean jointly Gaussian random variables in Problem 5.110. Find a transformation $A$ such that $\mathbf{Z}=A \mathbf{X}$ has components that are zero-mean, unitvariance Gaussian random variables.

Victor Salazar

Numerade Educator

Problem 120

In Example 5.47, suppose we estimate the value of the signal $X$ from the noisy observation $Y$ by:

$$
\hat{X}=\frac{1}{1+\sigma_N^2 / \sigma_X^2} Y .
$$

(a) Evaluate the mean square estimation error: $E\left[(X-\hat{X})^2\right]$.
(b) How does the estimation error in part a vary with signal-to-noise ratio $\sigma_X / \sigma_N$ ?

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10:07

Problem 121

Find the inverse of the cdf of the Rayleigh random variable to derive the transformation method for generating Rayleigh random variables. Show that this method leads to the same algorithm that was presented in Section 5.10.

Michael Twiton

Numerade Educator

Problem 122

Reproduce the results presented in Example 5.49.

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Problem 123

Consider the two-dimensional modem in Problem 5.36.
(a) Generate 10,000 discrete random variables uniformly distributed in the set $\{1,2,3,4\}$. Assign each outcome in this set to one of the signals $\{(1,1),(1,-1),(-1,1),(-1,-1)\}$. The sequence of discrete random variables then produces a sequence of 10,000 signal points $\mathbf{X}$.
(b) Generate 10,000 noise pairs $\mathbf{N}$ of independent zero-mean, unit-variance jointly Gaussian random variables.
(c) Form the sequence of 10,000 received signals $\mathbf{Y}=\left(Y_1, Y_2\right)=\mathbf{X}+\mathbf{N}$.
(d) Plot the scattergram of received signal vectors. Is the plot what you expected?
(e) Estimate the transmitted signal by the quadrant that $\mathbf{Y}$ falls in: $\hat{X}=\left(\operatorname{sgn}\left(Y_1\right)\right.$, $\left.\operatorname{sgn}\left(Y_2\right)\right)$.
(f) Compare the estimates with the actually transmitted signals to estimate the probability of error.

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Problem 124

Generate a sequence of 1000 pairs of independent zero-mean Gaussian random variables, where $X$ has variance 2 and $N$ has variance 1. Let $Y=X+N$ be the noisy signal from Example 5.47.
(a) Estimate $X$ using the estimator in Problem 5.120, and calculate the sequence of estimation errors.
(b) What is the pdf of the estimation error?
(c) Compare the mean, variance, and relative frequencies of the estimation error with the result from part b.

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Problem 125

Let $X_1, X_2, \ldots, X_{1000}$ be a sequence of zero-mean, unit-variance independent Gaussian random variables. Suppose that the sequence is "smoothed" as follows:

$$
Y_n=\left(X_n+X_{N-1}\right) / 2 \text { where } X_0=0 .
$$

(a) Find the pdf of $\left(Y_n, Y_{n+1}\right)$.
(b) Generate the sequence of $X_n$ and the corresponding sequence $Y_n$. Plot the scattergram of $\left(Y_n, Y_{n+1}\right)$. Does it agree with the result from part a?
(c) Repeat parts a and b for $Z_n=\left(X_n-X_{N-1}\right) / 2$.

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Problem 126

Let $X$ and $Y$ be independent, zero-mean, unit-variance Gaussian random variables. Find the linear transformation to generate jointly Gaussian random variables with means $m_1, m_2$, variances $\sigma_1^2, \sigma_2^2$, and correlation coefficient $\rho$. Hint: Use the conditional pdf in Eq. (5.64).

Victor Salazar

Numerade Educator

Problem 127

(a) Use the method developed in Problem 5.126 to generate 1000 pairs of jointly Gaussian random variables with $m_1=1, m_2=-1$, variances $\sigma_1^2=1, \sigma_2^2=2$, and correlation coefficient $\rho=-1 / 2$.
(b) Plot a two-dimensional scattergram of the 1000 pairs and compare to equal-pdf contour lines for the theoretical pdf.

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10:11

Problem 128

Let $H$ and $W$ be the height and weight of adult males. Studies have shown that $H$ (in cm) and $V=\ln W(W$ in kg$)$ are jointly Gaussian with parameters $m_H=174 \mathrm{~cm}, m_V=4.4$, $\sigma_H^2=42.36, \sigma_V^2=0.021$, and $\operatorname{COV}(H, V)=0.458$.
(a) Use the method in part a to generate 1000 pairs $(H, V)$. Plot a scattergram to check the joint pdf.
(b) Convert the $(H, V)$ pairs into $(H, W)$ pairs.
(c) Calculate the body mass index for each outcome, and estimate the proportion of the population that is underweight, normal, overweight, or obese. (See Problem 5.6.)

Chris Trentman

Numerade Educator

Problem 129

The random variables $X$ and $Y$ have joint pdf:

$$
f_{X, Y}(x, y)=c \sin (x+y) \quad 0 \leq x \leq \pi / 2,0 \leq y \leq \pi / 2
$$

(a) Find the value of the constant $c$.
(b) Find the joint cdf of $X$ and $Y$.
(c) Find the marginal pdf's of $X$ and of $Y$.
(d) Find the mean, variance, and covariance of $X$ and $Y$.

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04:25

Problem 130

An inspector selects an item for inspection according to the outcome of a coin flip: The item is inspected if the outcome is heads. Suppose that the time between item arrivals is an exponential random variable with mean one. Assume the time to inspect an item is a constant value $t$.
(a) Find the pmf for the number of item arrivals between consecutive inspections.
(b) Find the pdf for the time $X$ between item inspections. Hint: Use conditional expectation.
(c) Find the value of $p$, so that with a probability of $90 \%$ an inspection is completed before the next item is selected for inspection.

Ahmad Reda

Numerade Educator

05:29

Problem 131

The lifetime $X$ of a device is an exponential random variable with mean $=1 / R$. Suppose that due to irregularities in the production process, the parameter $R$ is random and has a gamma distribution.
(a) Find the joint pdf of $X$ and $R$.
(b) Find the pdf of $X$.
(c) Find the mean and variance of $X$.

Ahmad Reda

Numerade Educator

Problem 132

Let $X$ and $Y$ be samples of a random signal at two time instants. Suppose that $X$ and $Y$ are independent zero-mean Gaussian random variables with the same variance. When signal " 0 " is present the variance is $\sigma_0^2$, and when signal " 1 " is present the variance is $\sigma_1^2>\sigma_0^2$. Suppose signals 0 and 1 occur with probabilities $p$ and $1-p$, respectively. Let $R^2=X^2+Y^2$ be the total energy of the two observations.
(a) Find the pdf of $R^2$ when signal 0 is present; when signal 1 is present. Find the pdf of $R^2$.
(b) Suppose we use the following "signal detection" rule: If $R^2>T$, then we decide signal 1 is present; otherwise, we decide signal 0 is present. Find an expression for the probability of error in terms of $T$.
(c) Find the value of $T$ that minimizes the probability of error.

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Problem 133

Let $U_0, U_1, \ldots$ be a sequence of independent zero-mean, unit-variance Gaussian random variables. A "low-pass filter" takes the sequence $U_i$ and poloduces the output sequence $X_n=\left(U_n+U_{n-1}\right) / 2$, and a "high-pass filter" produces the output sequence $Y_n=\left(U_n-U_{n-1}\right) / 2$.
(a) Find the joint pdf of $X_n$ and $X_{n-1}$; of $X_n$ and $X_{n+m}, m>1$.
(b) Repeat part a for $Y_n$.
(c) Find the joint pdf of $X_n$ and $Y_m$.

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