Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia

Chapter 7 Sums of Random Variables and Long-Term Averages - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $Z=X+Y+Z$, where $X, Y$, and $Z$ are zero-mean, unit-variance random variables with $\operatorname{COV}(X, Y)=1 / 2$, and $\operatorname{COV}(Y, Z)=-1 / 4$ and $\operatorname{COV}(X, Z)=1 / 2$.
(a) Find the mean and variance of $Z$.
(b) Repeat part a assuming $X, Y$, and $Z$ are uncorrelated random variables.

Sheryl Ezze

Numerade Educator

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Problem 2

Let $X_1, \ldots, X_n$ be random variables with the same mean and with covariance function:

$$
\operatorname{COv}\left(X_i, X_j\right)= \begin{cases}\sigma^2 & \text { if } i=j \\ \rho \sigma^2 & \text { if }|i-j|=1 \\ 0 & \text { otherwise }\end{cases}
$$

where $|\rho|<1$. Find the mean and variance of $S_n=X_1+\cdots+X_n$.

Rashmi Sinha

Numerade Educator

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Problem 3

Let $X_1, \ldots, X_n$ be random variables with the same mean and with covariance function

$$
\operatorname{COv}\left(X_i, X_j\right)=\sigma^2 \rho^{i-j}
$$

where $|\rho|<1$. Find the mean and variance of $S_n=X_1+\cdots+X_n$.

Rashmi Sinha

Numerade Educator

05:50

Problem 4

Let $X$ and $Y$ be independent Cauchy random variables with parameters 1 and 4, respectively. Let $Z=X+Y$.
(a) Find the characteristic function of $Z$.
(b) Find the pdf of $Z$ from the characteristic function found in part a.

Tatiana Graham

Numerade Educator

01:52

Problem 5

Let $S_k=X_1+\cdots+X_k$, where the $X_i$ 's are independent random variables, with $X_i$ a chi-square random variable with $n_i$ degrees of freedom. Show that $S_k$ is a chi-square random variable with $n=n_1+\cdots+n_k$ degrees of freedom.

Manik Pulyani

Numerade Educator

Problem 6

Let $S_n=X_1^2+\cdots+X_n^2$, where the $X_i$ 's are iid zero-mean, unit-variance Gaussian random variables.
(a) Show that $S_n$ is a chi-square random variable with $n$ degrees of freedom. Hint: See Example 4.34.
(b) Use the methods of Section 4.5 to find the pdf of

$$
T_n=\sqrt{X_1^2+\cdots+X_n^2}
$$
(c) Show that $T_2$ is a Rayleigh random variable.
(d) Find the pdf for $T_3$. The random variable $T_3$ is used to model the speed of molecules in a gas. $T_3$ is said to have the Maxwell distribution.

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Problem 7

Let $X$ and $Y$ be independent exponential random variables with parameters 2 and 10 , respectively. Let $Z=X+Y$.
(a) Find the characteristic function of $Z$.
(b) Find the pdf of $Z$ from the characteristic function found in part a.

Victor Salazar

Numerade Educator

Problem 8

Let $Z=3 X-7 Y$, where $X$ and $Y$ are independent random variables.
(a) Find the characteristic function of $Z$.
(b) Find the mean and variance of $Z$ by taking derivatives of the characteristic function found in part a.

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Problem 9

Let $M_n$ be the sample mean of $n$ iid random variables $X_j$. Find the characteristic function of $M_n$ in terms of the characteristic function of the $X_i$ 's.

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Problem 10

The number $X_j$ of raffle winners in classroom $j$ is a binomial random variable with parameter $n_j$ and $p$. Suppose that the school has $K$ classrooms. Find the pmf of the total number of raffle winners in the school, assuming the $X_i$ 's are independent random variables.

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Problem 11

The number of packet arrivals $X_i$ at port $i$ in a router is a Poisson random variable with mean $\alpha_i$. Given that the router has $k$ ports, find the pmf for the total number of packet arrivals at the router. Assume that the $X_i$ 's are independent random variables.

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Problem 12

Let $X_1, X_2, \ldots$ be a sequence of independent integer-valued random variables, let $N$ be an integer-valued random variable independent of the $X_j$, and let

$$
S=\sum_{k=1}^N X_k
$$

(a) Find the mean and variance of $S$.
(b) Show that

$$
G_S(z)=E\left(z^S\right)=G_N\left(G_X(z)\right)
$$

where $G_X(z)$ is the generating function of each of the $X_k$ 's.

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05:51

Problem 13

Let the number of smasher ef the $X_k$ 's. dom variable with mean $L$-up cars arriving at a body shop in a week be a Poisson ranables that are equally like. Each job repair costs $X_j$ dollars, the $X_j$ 's are iid random vari-
(a) Find the mean and to be $$\$ 500$$ or $$\$ 1000$$.
(b) Find the $G_R(z)=E\left[z^R\right]$.

Rosina Dapaah

Numerade Educator

03:18

Problem 14

Let the number of widgets tested in an assembly line in 1 hour be a binomial random variable with parameters $n=600$ and $p$. Suppose that the probability that a widget is faulty is $a$. Let $S$ be the number of widgets that are found faulty in a 1-hour period.
(a) Find the mean and variance of $S$.
(b) Find $G_S(z)=E\left[z^s\right]$.

Stanley Enemuo

Numerade Educator

02:40

Problem 15

Suppose that the number of particle emissions by a radioactive mass in $t$ seconds is a Poisson random variable with mean $\lambda t$. Use the Chebyshev inequality to obtain a bound for the probability that $|N(t) / t-\lambda|$ exceeds $\varepsilon$.

Sheryl Ezze

Numerade Educator

00:50

Problem 16

Suppose that $20 \%$ of voters are in favor of certain legislation. A large number $n$ of voters are polled and a relative frequency estimate $f_A(n)$ for the above proportion is obtained.
Use Eq. (7.20) to determine how many voters should be polled in order that the probability is at least .95 that $f_A(n)$ differs from 0.20 by less than 0.02 .

Hast Aggarwal

Numerade Educator

01:58

Problem 17

A fair die is tossed 20 times. Use Eq. (7.20) to bound the probability that the total number of dots is between 60 and 80 .

Dharmendra Jain

Numerade Educator

01:10

Problem 18

Let $X_i$ be a sequence of independent zero-mean, unit-variance Gaussian random variables. Compare the bound given by Eq. (7.20) with the exact value obtained from the $Q$ function for $n=16$ and $n=81$.

Ameer Said

Numerade Educator

Problem 19

Does the weak law of large numbers hold for the sample mean if the $X_i$ 's have the covariance functions given in Problem 7.2? Assume the $X_i$ have the same mean.

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Problem 20

Repeat Problem 7.19 if the $X_i$ 's have the covariance functions given in Problem 7.3.

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27:31

Problem 21

(The sample variance) Let $X_1, \ldots, X_n$ be an id sequence of random variables for which the mean and variance are unknown. The sample variance is defined as follows:

$$
V_n^2=\frac{1}{n-1} \sum_{j=1}^n\left(X_j-M_n\right)^2
$$

where $M_n$ is the sample mean.
(a) Show that

$$
\sum_{j=1}^n\left(X_j-\mu\right)^2=\sum_{j=1}^n\left(X_j-M_n\right)^2+n\left(M_n-\mu\right)^2
$$

(b) Use the result in part a to show that

$$
E\left[k \sum_{j=1}^n\left(X_j-M_n\right)^2\right]=k(n-1) \sigma^2
$$

(c) Use part b to show that $E\left[V_n^2\right]=\sigma^2$. Thus $V_n^2$ is an unbiased estimator for the variance.
(d) Find the expected value of the sample variance if $n-1$ is replaced by $n$. Note that this is a biased estimator for the variance.

Jeremiah Mbaria

Numerade Educator

Problem 22

(a) A fair coin is tossed 100 times. Estimate the probability that the number of heads is between 40 and 60 . Estimate the probability that the number is between 50 and 55 .
(b) Repeat part a for $n=1000$ and the intervals $[400,600]$ and $[500,550]$.

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01:42

Problem 23

Repeat Problem 7.16 using the central limit theorem.

Aman Gupta

Numerade Educator

01:55

Problem 24

Use the central limit theorem to estimate the probability in Problem 7.17 .

Amany Waheeb

Numerade Educator

02:19

Problem 25

The lifetime of a cheap light bulb is an exponential random variable with mean 36 hours. Suppose that 16 light bulbs are tested and their lifetimes measured. Use the central limit theorem to estimate the probability that the sum of the lifetimes is less than 600 hours.

Clarissa Noh

Numerade Educator

Problem 26

A student uses pens whose lifetime is an exponential random variable with mean 1 week. Use the central limit theorem to determine the minimum number of pens he should buty at the beginning of a 15 -week semester, so that with probability .99 he does not run out $d$ pens during the semester.

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04:41

Problem 27

Let $S$ be the sum of 80 iid Poisson random variables with mean 0.25 . Compare the exact value of $P[S=k]$ to an approximation given by the central limit theorem as Eq. (7.30).

Mengchun Cai

Numerade Educator

06:55

Problem 28

The number of messages arriving at a multiplexer is a Poisson random variable with mean 15 messages/second. Use the central limit theorem to estimate the probability that more than 950 messages arrive in one minute.

Sonam Khatri

Numerade Educator

Problem 29

A binary transmission channel introduces bit errors with probability .15 . Estimate the probability that there are 20 or fewer errors in 100 bit transmissions.

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Problem 30

The sum of a list of 100 real numbers is to be computed. Suppose that numbers are rounded off to the nearest integer so that each number has an error that is uniformly distributed in the interval $(-0.5,0.5)$. Use the central limit theorem to estimate the probability that the total error in the sum of the 64 numbers exceeds 4.

Rashmi Sinha

Numerade Educator

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Problem 31

(a) A fair coin is tossed 100 times. Use the Chernoff bound to estimate the probability that the number of heads is greater than 90 . Compare to an estimate using the central limit theorem.
(b) Repeat part a for $n=1000$ and the probability that the number of heads is greater than 650.

Rashmi Sinha

Numerade Educator

10:39

Problem 32

A binary transmission channel introduces bit errors with probability .01 . Use the Chernoff bound to estimate the probability that there are more than 3 errors in 100 bit transmissions. Compare to an estimate using the central limit theorem.

Chris Trentman

Numerade Educator

06:31

Problem 33

(a) When you play the rock/paper/scissors game against your sister you lose with probability $3 / 5$. Use the Chernoff bound to estimate the probability that you win more than half of 20 games played.
(b) Repeat for 100 games.
(c) Use trial and error to find the number of games $n$ that need to be played so that the probability that your sister wins more than $1 / 2$ the games is $90 \%$.

Heena Haldankar

Numerade Educator

Problem 34

Show that the Chernoff bound for $X$, a Poisson random variable with mean $\alpha$, is $P[X \geq a] \leq e^{-a \ln (a / \alpha)+a-\alpha}$ for $a>\alpha$. Hint: Use $E\left[e^{s X}\right]=e^{\alpha\left(e^s-1\right)}$.

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Problem 35

Redo Problem 7.26 using the Chernoff bound.

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03:23

Problem 36

Show that the Chernoff bound for $X$, a Gaussian random variable with mean $\mu$ and variance $\sigma^2$, is $P[X \geq a] \leq e^{-(a-\mu)^2 / 2 \sigma^2}, a>\mu$. Hint: Use $E\left[e^{s X}\right]=e^{s \mu+s^2 \sigma^2 / 2}$.

Sagar Singh

Numerade Educator

Problem 37

Compare the Chernoff bound for the Gaussian random variable with the estimates provided by Eq. (4.54).

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Problem 38

(a) Find the Chernoff bound for the exponential random variable with rate $\lambda$.
(b) Compare the exact probability of $P[X \geq k / \lambda]$ with the Chernoff bound.

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01:32

Problem 39

(a) Generalize the approach in Problem 7.38 to find the Chernoff bound for a gamma random variable with parameters $\lambda$ and $\alpha$.
(b) Use the result of part a to obtain the Chernoff bound for a chi-square random variable with $k$ degrees of freedom.

Manik Pulyani

Numerade Educator

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Problem 40

Let $U_n(\zeta), W_n(\zeta), Y_n(\zeta)$, and $Z_n(\zeta)$ be the sequences of random variables defined in Example 7.18.
(a) Plot the sequence of functions of $\zeta$ associated with each sequence of random variables.
(b) For $\zeta=1 / 4$, plot the associated sample sequence.

Shu Naito

Numerade Educator

Problem 41

Let $\zeta$ be selected at random from the interval $S=[0,1]$, and let the probability that $\zeta$ is in a subinterval of $S$ be given by the length of the subinterval. Define the following sequences of random variables for $n \geq 1$ :

$$
X_n(\zeta)=\zeta^n, Y_n(\zeta)=\cos ^2 2 \pi \zeta, Z_n(\zeta)=\cos ^n 2 \pi \zeta
$$
Do the sequences converge, and if so, in what sense and to what limiting random variable?

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Problem 42

Let $b_i, i \geq 1$, be a sequence of iid, equiprobable Bernoulli random variables, and let $\zeta$ be the number between $[0,1]$ determined by the binary expansion

$$
\zeta=\sum_{i=1}^{\infty} b_i 2^{-i}
$$

(a) Explain why $\zeta$ is uniformly distributed in $[0,1]$.
(b) How would you use this definition of $\zeta$ to generate the sample sequences that occur in the urn problem of Example 7.20?

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05:33

Problem 43

Let $X_n$ be a sequence of iid, equiprobable Bernoulli random variables, and let

$$
Y_n=2^n X_1 X_2 \ldots X_n
$$

(a) Plot a sample sequence. Does this sequence converge almost surely, and if so, to what limit?
(b) Does this sequence converge in the mean square sense?

James Kiss

Numerade Educator

Problem 44

Let $X_n$ be a sequence of iid random variables with mean $m$ and variance $\sigma^2<\infty$. Let $M_n$ be the associated sequence of arithmetic averages,

$$
M_n=\frac{1}{n} \sum_{i=0}^n X_i
$$

Show that $M_n$ converges to $m$ in the mean square sense.

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Problem 45

Let $X_n$ and $Y_n$ be two (possibly dependent) sequences of random variables that converge in the mean square sense to $X$ and $Y$, respectively. Does the sequence $X_n+Y_n$ converge in the mean square sense, and if so, to what limit?

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Problem 46

Let $U_n$ be a sequence of iid zero-mean, unit-variance Gaussian random variables. A "lowpass filter" takes the sequence $U_n$ and produces the sequence

$$
X_n=\frac{1}{2}\left(U_n+U_{n-1}\right) .
$$

(a) Does this sequence converge in the mean square sense?
(b) Does it converge in distribution?

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Problem 47

Does the sequence of random variables introduced in Example 7.20 converge in the mean square sense?

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Problem 48

Customers arrive at an automated teller machine at discrete instants of time, $n=1,2, \ldots$ The number of customer arrivals in a time instant is a Bernoulli random variable with pa rameter $p$, and the sequence of arrivals is iid. Assume the machine services a customer in less than one time unit. Let $X_n$ be the total number of customers served by the machine up to time $n$. Suppose that the machine fails at time $N$, where $N$ is a geometric random variable with mean 100 , so that the customer count remains at $X_N$ thereafter.
(a) Sketch a sample sequence for $X_n$.
(b) Do the sample sequences converge almost surely, and if so, to what limit?
(c) Do the sample sequences converge in the mean square sense?

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Problem 49

Show that the sequence $Y_n(\zeta)$ defined in Example 7.18 converges in distribution.

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Problem 50

Let $X_n$ be a sequence of Laplacian random variables with parameter $\alpha=n$. Does this se quence converge in distribution?

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02:05

Problem 51

The customer arrival times at a bus depot are iid exponential random variables with mean 1 minute. Suppose that buses leave as soon as 30 seats are full. At what rate do buses leave the depot?

Amany Waheeb

Numerade Educator

00:20

Problem 52

A faulty clock ticks forward every minute with probability $p=0.1$ and it does not tick forward with probability $1-p$. What is the rate at which this clock moves forward?

Amrita Bhasin

Numerade Educator

Problem 53

(a) Show that $\{N(t) \geq n\}$ and $\left\{S_n \leq t\right\}$ are equivalent events.
(b) Use part a to find $P[N(t) \leq n]$ when the $X_i$ are iid exponential random variables with mean $1 / \alpha$.

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Problem 54

Explain why the following are not equivalent events:
(a) $\{N(t) \leq n\}$ and $\left\{S_n \geq t\right\}$.
(b) $\{N(t)>n\}$ and $\left\{S_n<t\right\}$.

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01:56

Problem 55

A communication channel alternates between periods when it is error free and periods during which it introduces errors. Assuming that these periods are independent random variables of means $m_1=100$ hours and $m_2=1$ minute, respectively, find the long-term proportion of time during which the channel is error free.

Hast Aggarwal

Numerade Educator

Problem 56

A worker works at a rate $r_1$ when the boss is around and at a rate $r_2$ when the boss is not present. Suppose that the sequence of durations of the time periods when the boss is present and absent are independent random variables with means $m_1$ and $m_2$, respectively. Find the long-term average rate at which the worker works.

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02:04

Problem 57

A computer (repairman) continuously cycles through three tasks (machines). Suppose that each time the computer services task $i$, it spends time $X_i$ doing so.
(a) What is the long-term rate at which the computer cycles through the three tasks?
(b) What is the long-term proportion of time spent by the computer servicing task $i$ ?
(c) Repeat parts a and b if a random time $W$ is required for the computer (repairman) to switch (walk) from one task (machine) to another.

Hossam Mohamed

Numerade Educator

04:44

Problem 58

Customers arrive at a phone booth and use the phone for a random time $Y$, with mean 3 minutes, if the phone is free. If the phone is not free, the customers leave immediately. Suppose that the time between customer arrivals is an exponential random variable with mean 10 minutes.
(a) Find the long-term rate at which customers use the phone.
(b) Find the long-term proportion of customers that leave without using the phone.

Robin Corrigan

Numerade Educator

03:16

Problem 59

The lifetime of a certain system component is an exponential random variable with mean $T=2$ months. Suppose that the component is replaced when it fails or when it reaches the age of $3 T$ months.
(a) Find the long-term rate at which components are replaced.
(b) Find the long-term rate at which working components are replaced.

Linh Vu

Numerade Educator

Problem 60

A data compression encoder segments a stream of information bits into patterns as shown below. Each pattern is then encoded into the codeword shown below.
$$
\begin{array}{lll}
\hline \text { Pattern } & \text { Codeword } & \text { Probability } \\
\hline 1 & 100 & .1 \\
01 & 101 & .09 \\
001 & 110 & .081 \\
0001 & 111 & .0729 \\
0000 & 0 & .6521 \\
\hline
\end{array}
$$
(a) If the information source produces a bit every millisecond, find the rate at which codewords are produced.
(b) Find the long-term ratio of encoded bits to information bits.

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Problem 61

In Example 7.29 evaluate the proportion of time that the residual lifetime $r(t)$ exceeds $c$ seconds for the following cases:
(a) $X_j$ iid uniform random variables in the interval $[0,2]$.
(b) $X_i$ iid exponential random variables with mean 1.
(c) $X_j$ iid Rayleigh random variables with mean 1.
(d) Calculate and compare the mean residual time in each of the above three cases.

Victor Salazar

Numerade Educator

Problem 62

Let the age $a(t)$ of a cycle be defined as the time that has elapsed from the last arrival up to an arbitrary time instant $t$. Show that the long-term proportion of time that $a(t)$ exceeds $c$ seconds is given by Eq. (7.48).

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Problem 63

Suppose that the cost in each cycle grows at a rate proportional to the age $a(t)$ of the cycle, that is,

$$
C_j=\int_0^{X_t} a\left(t^{\prime}\right) d t^{\prime}
$$

(a) Show that $C_j=X_j^2 / 2$.
(b) Show that the long-term rate at which the cost grows is $E\left[X^2\right] / 2 E[X]$.
(c) Show that the result in part b is also the long-term time average of $a(t)$, that is,

$$
\lim _{t \rightarrow \infty} \frac{1}{t} \int_0^t a\left(t^{\prime}\right) d t^{\prime}=\frac{E\left[X^2\right]}{2 E[X]}
$$

(d) Explain why the average residual life is also given by the above expression.

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01:46

Problem 64

Calculate the mean age and mean residual life in Problem 7.63 in the following cases:
(a) $X_j$ iid uniform random variables in the interval $[0,2]$.
(b) $X_j$ iid exponential random variables with mean 1.
(c) $X_i$ iid Rayleigh random variables with mean 1.

Amany Waheeb

Numerade Educator

Problem 65

(The Regenerative Method) Suppose that a queueing system has the property that when a customer arrives and finds an empty system, the future behavior of the system is completely independent of the past. Define a cycle to consist of the time period between two consecutive customer arrivals to an empty system. Let $N_j$ be the number of customers served during the $j$ th cycle and let $T_j$ be the total delay of all customers served during the $j$ th cycle.
(a) Use Theorem 2 to show that the average customer delay is given by $E[T] / E[N]$, that is,

$$
\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n D_k=\frac{E[T]}{E[N]},
$$

where $D_k$ is the delay of the $k$ th customer.
(b) How would you use this result to estimate the average delay in a computer simulation of a queueing system?

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Problem 66

Let the discrete random variable $X$ be uniformly distributed in the set $\{0,1,2\}$.
(a) Find the $N=3$ DFT for $X$.
(b) Use the inverse DFT to recover $P[X=1]$.

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10:31

Problem 67

Let $S=X+Y$, where $X$ and $Y$ are iid random variables uniformly distributed in the set $\{0,1,2\}$.
(a) Find the $N=5 \mathrm{DFT}$ for $S$.
(b) Use the inverse DFT to find $P[S=2]$.

Mengchun Cai

Numerade Educator

Problem 68

Let $X$ be a binomial random variable with parameter $n=8$ and $p=1 / 2$.
(a) Use the FFT to obtain the pmf of $X$ from $\Phi_X(\omega)$.
(b) Use the FFT to obtain the pmf of $Z=X+Y$ where $X$ and $Y$ are iid binomial random variables with $n=8$ and $p=1 / 2$.

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Problem 69

Let $X_i$ be a discrete random variable that is uniformly distributed in the set $\{0,1, \ldots, 9\}$. Use the FFT to find the pmf of $S_n=X_i+\cdots+X_n$ for $n=5$ and $n=10$. Plot your results and compare them to Fig. 7.16.

Shu Naito

Numerade Educator

08:36

Problem 70

Let $X$ be the geometric random variable with parameter $p=1 / 2$. Use the FFT to evaluate Eq. (7.55) to compute $p_k^{\prime}$ for $N=8$ and $N=16$. Compare the results to those given by Eq. (7.57).

Robin Corrigan

Numerade Educator

04:24

Problem 71

Let $X$ be a Poisson random variable with mean $L=5$.
(a) Use the FFT to obtain the pmf from $\Phi_X(\omega)$. Find the value of $N$ for which the error in Eq. (7.55) is less than $1 \%$.
(b) Let $S=X_1+X_2+\cdots+X_5$, where the $X_i$ are iid Poisson random variables with mean $L=5$. Use the FFT to compute the pmf of $S$ from $\Phi_X(\omega)$.

Barsha Rana

Numerade Educator

Problem 72

The probability generating function for the number $N$ of customers in a certain queueing system (the so-called M/D/1 system discussed in Chapter 12) is

$$
G_N(z)=\frac{(1-\rho)(1-z)}{1-z e^{\rho(1-z)}}
$$

where $0 \leq \rho \leq 1$. Use the FFT to obtain the pmf of $N$ for $\rho=1 / 2$.

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01:45

Problem 73

Use the FFT to obtain approximately the pdf of a Laplacian random variable from its characteristic function. Use the same parameters as in Example 7.33 and compare your results to those shown in Fig. 7.17.

Manik Pulyani

Numerade Educator

Problem 74

Use the FFT to obtain approximately the pdf of $Z=X+Y$, where $X$ and $Y$ are independent Laplacian random variables with parameters $\alpha=1$ and $\alpha=2$, respectively.

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Problem 75

Use the FFT to obtain approximately the pdf of a zero-mean, unit-variance Gaussian random variable from its characteristic function. Experiment with the values of $N$ and $\omega_0$ and compare the results given by the FFT with the exact values.

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01:01

Problem 76

Figures 7.2 through 7.4 for the cdf of the sum of iid Bernoulli, uniform, and exponential random variables were obtained using the FFT. Reproduce the results shown in these figures.

Manik Pulyani

Numerade Educator

01:31

Problem 77

The number $X$ of type 1 defects in a system is a binomial random variable with parameters $n$ and $p$, and the number $Y$ of type 2 defects is binomial with parameters $m$ and $r$.
(a) Find the probability generating function for the total number of defects in the system.
(b) Find an expression for the probability that the total number of defects is $k$.
(c) Let $n=32, p=1 / 10$, and $m=16, r=1 / 8$. Use the FFT to evaluate the pmf for the total number of defects in the system.

Christopher Stanley

Numerade Educator

Problem 78

Let $U_n$ be a sequence of iid zero-mean, unit-variance Gaussian random variables. A "lowpass filter" takes the sequence $U_n$ and produces the sequence

$$
X_n=\frac{1}{2} U_n+\left(\frac{1}{2}\right)^2 U_{n-1}+\cdots+\left(\frac{1}{2}\right)^n U_1
$$
(a) Find the mean and variance of $X_n$.
(b) Find the characteristic function of $X_n$. What happens as $n$ approaches infinity?
(c) Does this sequence of random variables converge? In what sense?

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Problem 79

Let $S_n$ be the sum of a sequence of $X_i$ 's that are jointly Gaussian random variables with mean $\mu$ and with the covariance function given in Problem 7.2.
(a) Find the characteristic function of $S_n$.
(b) Find the mean and variance of $S_n-S_m$.
(c) Find the joint characteristic function of $S_n$ and $S_m$. Hint: Assuming $n>m$, condition on the value of $S_m$.
(d) Does $S_n$ converge in the mean square sense?

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Problem 80

Repeat Problem 7.79 with the sequence of $X_i$ 's given as jointly Gaussian random variables with mean and covariance functions given in Problem 7.3.

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Problem 81

Let $Z_n$ be the sequence of random variables defined in the formulation of the centra! limit theorem, Eq. (7.26a). Does $Z_n$ converge in the mean square sense?

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Problem 82

Let $X_n$ be the sequence of independent, identically distributed outputs of an information source. At time $n$, the source produces symbols according to the following probabilities:
$$
\begin{array}{ccc}
\hline \text { Symbol } & \text { Probability } & \text { Codeword } \\
\hline \text { A } & 1 / 2 & 0 \\
\text { B } & 1 / 4 & 10 \\
\text { C } & 1 / 8 & 110 \\
\text { D } & 1 / 16 & 1110 \\
\text { E } & 1 / 16 & 1111 \\
\hline
\end{array}
$$
(a) The self-information of the output at time $n$ is defined by the random variable $Y_n=-\log _2 P\left[X_n\right]$. Thus, for example, if the output is $C$, the self-information is $-\log _2 1 / 8=3$. Find the mean and variance of $Y_n$. Note that the expected value of the self-information is equal to the entropy of $X$ (cf. Section 4.10).
(b) Consider the sequence of arithmetic averages of the self-information:

$$
S_n=\frac{1}{n} \sum_{k=1}^n Y_k
$$

Do the weak law and strong law of large numbers apply to $S_n$ ?
(c) Now suppose that the outputs of the information source are encoded using the vari-able-length binary codewords indicated above. Note that the length of the codewords corresponds to the self-information of the corresponding symbol. Interpret the result of part $b$ in terms of the rate at which bits are produced when the above code is applied to the information source outputs.

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