# Home work 3

Topics: Probability theory, Random variable, Cumulative distribution function Pages: 7 (816 words) Published: September 26, 2014
﻿Case Weatherhead School of ManagementOPRE 433 Fall 2014

Homework 3 – SOLUTION

Answer the following questions using JMP wherever you can.

1. Screening at an airport occurs at locations A, B, and C. A handles 50% of the passengers, B handles 30%, and C handles 20%. The detection rates for prohibited items (such as weapons) at the three locations are 0.9, 0.8, and 0.85, respectively. A. If a passenger at a boarding gate is found with a prohibited item, what is the probability that the passenger was screened at A? Suppose:

A: Item goes by screening location A
B: Item goes by screening location B
C: Item goes by screening location C
D: Prohibited item is detected at a screening location
So P(A)=0.5, P(B)=0.3, P(C)=0.2, P(D|A)=0.9, P(D|B)=0.8, P(D|C)=0.85 According to Bayes’ Rule:

So if a passenger at a boarding gate is found with a prohibited item, the probability that the passenger was screened at A is 0.523.

B. If a passenger at a boarding gate is found with a prohibited item, what is the probability that the passenger was screened at B? According to Bayes’ Rule:

So if a passenger at a boarding gate is found with a prohibited item, the probability that the passenger was screened at B is 0.279.

2. Let X equal the number observed on the throw of a single balanced die. A. Graph the probability mass function of X. (See the hint after the last question in this homework assignment). The probability mass function of X can be drawn in JMP as follows:

B. What is
According to the JMP output,

C. What is the standard deviation of X?
As shown in the JMP output,

D. Locate the interval on the abscissa (x-axis) of the graph in part A. What proportion of all values of X fall in this range? The interval is equivalent to the interval (0.084, 6.916), which means all values of X are located in this range.

3. A piece of electronic equipment has six computer chips. Although two of them are defective, it isn’t known which two of the six are defective. Three chips are selected at random, removed from the equipment, and inspected. Let X be the number of defective chips that are found in the inspection, where X = 0, 1, or 2. A. Find the probability mass function of X.

p(0) = 0.2, p(1) = 0.6, and p(2) = 0.2

B. Graph the probability mass function.

4. Let X be an r.v. (random variable) with the following probability distribution: p(k) = x2/a if k = -3, -2, -1, 0, 1, 2, 3 and p(k) = 0 for any other value of k. a. Find a.
Since different values of x represent mutually exclusive numerical events, summing p(x) over all values of x must equal 1:

b. Find E(X).

c. What is the probability distribution of the r.v. Y = [X – E(X)]2? Since E(X)=0, Y=[X-E(X)]2=X2
P(Y=0)=P(X=0)=0
P(Y=1)=P(X=-1)+P(X=1)=1/14
P(Y=4)=P(X=-2)+P(X=2)=2/7
P(Y=9)=P(X=-3)+P(X=3)=9/14
y
1
4
9
p(y)
1/14
2/7
9/14

d. Use part (c) to find the variance of X.
Var(X)=E{[X-E(X)]2}=E(Y)=1*1/14+4*2/7+9*9/14=98/14=7

e. Find the variance of X using the formula .

f. Find the variance of X using the formula .
Since Y=X2, E(X2)=E(Y)=7, and E(X)=0
So

5. Forty percent of Americans who travel by car look for gas stations and food outlets that are close to or visible from the highway. Suppose that a random sample of n = 25 travelers are asked how they determine where to stop for food and gas. Let X be the number in the sample who answer that they look for gas stations and food outlets that are close to or visible from the highway. A. What are the mean and variance of X?

Given the information, X has a binomial distribution because: (1) this experiment consists of n identical trials;
(2) each trial provides two outcomes, yes or no;
(3) the probability of yes (or no) on each trial remains the same; (4) trials are independent.
Using the formulas for the mean and variance of a binomial distribution, E(X) = np = 25*0.4 = 10
Var(X) = npq = 25*0.4*0.6=6...