Chapter 4, Continuous Random Variables and Probability Distributions Video Solutions, Applied Statistics and Probability for Engineers

Problem 1

Suppose that $f(x)=e^{-x}$ for $0<x .$ Determine the following:
(a) $P(1<X)$
(b) $P(1<X<2.5)$
(c) $P(X=3)$
(d) $P(X<4)$
(e) $P(3 \leq X)$
(f) $x$ such that $P(x<X)=0.10$
(g) $x$ such that $P(X \leq x)=0.10$

Robin Corrigan

Numerade Educator

04:48

Problem 2

Suppose that $f(x)=3\left(8 x-x^{2}\right) / 256$ for $0<x<8$ Determine the following:
(a) $P(X<2)$
(b) $P(X<9)$
(c) $P(2<X<4)$
(d) $P(X>6)$
(e) $x$ such that $P(X<x)=0.95$

Yingtai Xiao

Numerade Educator

03:50

Problem 3

Suppose that $f(x)=0.5 \cos x$ for $-\pi / 2<x<\pi / 2$. Determine the following:
(a) $P(X<0)$
(b) $P(X<-\pi / 4)$
(c) $P(-\pi / 4<X<\pi / 4)$
(d) $P(X>-\pi / 4)$
(e) $x$ such that $P(X<x)=0.95$

Yingtai Xiao

Numerade Educator

05:39

Problem 4

The diameter of a particle of contamination (in micrometers) is modeled with the probability density function $f(x)=2 / x^{3}$ for $x>1 .$ Determine the following:
(a) $P(X<2)$
(b) $P(X>5)$
(c) $P(4<X<8)$
(d) $P(X<4$ or $X>8)$
(e) $x$ such that $P(X<x)=0.95$

Mengchun Cai

Numerade Educator

03:02

Problem 5

Go Tutorial Suppose that $f(x)=x / 8$ for $3<x<5$. Determine the following probabilities:
(a) $P(X<4)$
(b) $P(X>3.5)$
(c) $P(4<X<5)$
(d) $P(X<4.5)$
(e) $P(X<3.5$ or $X>4.5)$

Yingtai Xiao

Numerade Educator

05:08

Problem 6

Suppose that $f(x)=e^{-(x-4)}$ for $4<x .$ Determine the following:
(a) $P(1<X)$
(b) $P(2 \leq X<5)$
(c) $P(5<X)$
(d) $P(8<X<12)$
(e) $x$ such that $P(X<x)=0.90$

Robin Corrigan

Numerade Educator

04:30

Problem 7

Suppose that $f(x)=1.5 x^{2}$ for $-1<x<1 .$ Determine the following:
(a) $P(0<X)$
(b) $P(0.5<X)$
(c) $P(-0.5 \leq X \leq 0.5)$
(d) $P(X<-2)$
(e) $P(X<0$ or $X>-0.5)$
(f) $x$ such that $P(x<X)=0.05$.

Yingtai Xiao

Numerade Educator

04:56

Problem 8

The probability density function of the time to failure of an electronic component in a copier (in hours) is $f(x)=$ $e^{-x / 1000 / 1000}$ for $x>0 .$ Determine the probability that
(a) A component lasts more than 3000 hours before failure.
(b) A component fails in the interval from 1000 to 2000 hours.
(c) A component fails before 1000 hours.
(d) The number of hours at which $10 \%$ of all components have failed

Amany Waheeb

Numerade Educator

01:29

Problem 9

The probability density function of the net weight in pounds of a packaged chemical herbicide is $f(x)=2.0$ for $49.75<x<50.25$ pounds.
(a) Determine the probability that a package weighs more than 50 pounds.
(b) How much chemical is contained in $90 \%$ of all packages?

Yingtai Xiao

Numerade Educator

02:28

Problem 10

The probability density function of the length of a cutting blade is $f(x)=1.25$ for $74.6<x<75.4$ millimeters. Determine the following:
(a) $P(X<74.8)$
(b) $P(X<74.8$ or $X>75.2)$
(c) If the specifications for this process are from 74.7 to 75.3

Yingtai Xiao

Numerade Educator

01:23

Problem 11

The probability density function of the length of a metal rod is $f(x)=2$ for $2.3<x<2.8$ meters.
(a) If the specifications for this process are from 2.25 to 2.75 meters, what proportion of rods fail to meet the specifications?
(b) Assume that the probability density function is $f(x)=2$ for an interval of length 0.5 meters. Over what value should the density be centered to achieve the greatest proportion of rods within specifications?

Yingtai Xiao

Numerade Educator

View

Problem 12

An article in Electric Power Systems Research ["Modeling Real-Time Balancing Power Demands in Wind Power Systems Using Stochastic Differential Equations" (2010, Vol. $80(8),$ pp. $966-974$ ) ] considered a new probabilistic model to balance power demand with large amounts of wind power. In this model, the power loss from shutdowns is assumed to have a triangular distribution with probability density function
$$
f(x)=\left\{\begin{array}{cc}
-5.56 \times 10^{-4}+5.56 \times 10^{-6} x, & x \in[100,500] \\
4.44 \times 10^{-3}-4.44 \times 10^{-6} x, & x \in[500,1000] \\
0, & \text { otherwise }
\end{array}\right.
$$
Determine the following:
(a) $P(X<90)$
(b) $P(100<X \leq 200)$
(c) $P(X>800)$
(d) Value exceeded with probability 0.1 .

Rashmi Sinha

Numerade Educator

03:02

Problem 13

A test instrument needs to be calibrated periodically to prevent measurement errors. After some time of use without calibration, it is known that the probability density function of the measurement error is $f(x)=1-0.5 x$ for $0<x<2$ millimeters.
(a) If the measurement error within 0.5 millimeters is acceptable, what is the probability that the error is not acceptable before calibration?
(b) What is the value of measurement error exceeded with probability 0.2 before calibration?
(c) What is the probability that the measurement error is exactly 0.22 millimeters before calibration?

Amany Waheeb

Numerade Educator

04:18

Problem 14

The distribution of $X$ is approximated with a triangular probability density function $f(x)=0.025 x-0.0375$ for $30<x<50$ and $f(x)=-0.025 x+0.0875$ for $50<x<70$
Determine the following:
(a) $P(X \leq 40)$
(b) $P(40<X \leq 60)$
(c) Value $x$ exceeded with probability 0.99 .

Amany Waheeb

Numerade Educator

02:17

Problem 15

The waiting time for service at a hospital emergency department (in hours) follows a distribution with probability density function $f(x)=0.5 \exp (-0.5 x)$ for $0<x$. Determine the following:
(a) $P(X<0.5)$
(b) $P(X>2)$
(c) Value $x$ (in hours) exceeded with probability 0.05 .

Yingtai Xiao

Numerade Educator

02:06

Problem 16

If $X$ is a continuous random variable, argue that $P\left(x_{1} \leq X \leq x_{2}\right)=P\left(x_{1}<X \leq x_{2}\right)=P\left(x_{1} \leq X<x_{2}\right)=P\left(x_{1}<X<x_{2}\right)$

Robin Corrigan

Numerade Educator