Math 3339Homework 4 (4.9, Chapter 5)

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1. For 0 ≤ 𝑥 ≤ 1, let 𝑓(𝑥) = 𝑘𝑥(1 − 𝑥), where k is a constant.

(a) Find the value of k such that f is a density function.

(b) What is the mean value of this random variable?

(c) What is the variance of this random variable?

(d) Construct the cumulative distribution for this random variable.

a

Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2

2.

a

for 4 ≤ 𝑥 ≤ 6 and 𝑓(𝑥) = 0 elsewhere.

(a) Check that the total area under the probability density function is equal to 1.

(b) What is 𝑃(4 ≤ 𝑥 ≤ 5)?

(c) Construct the cumulative distribution function.

(d) What is the expected value of this random variable?

(e) What is the variance of this random variable?

(f) What is the standard deviation of this random variable?

a

1

A random variable X takes values between 4 and 6 with a probability density function 𝑓(𝑥) = 𝑥𝑙𝑛(1.5)

Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2

3.

a

a

𝑥2

A random variable X takes values between 0 and 4 with a cumulative distribution function 𝐹(𝑥) = 16

for 0 ≤ 𝑥 ≤ 4.

(a) Sketch the cumulative distribution function

(b) What is 𝑃(𝑋 ≤ 2)?

(c) What is 𝑃(1 ≤ 𝑋 ≤ 3)?

(d) Construct and sketch the probability density function.

(e) What is the expected value of this random variable?

(f) What is the median of this random variable?

Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2

4. Suppose that 𝑋~𝑈𝑛𝑖𝑓𝑜𝑟𝑚( −2,8), find

(a) The mean value of this random variable.

(b) The standard deviation of this random variable.

(c) Find the 80th percentile of the distribution.

(d) 𝑃(1 ≤ 𝑋 ≤ 3)

5. Let X = the time in hours between two successive arrivals at the drive-up window of a fast food

restaurant. If X has an exponential distribution with 𝜆 = 10, compute the following:

(a) The expected time between two successive arrivals.

(b) The standard deviation of the time between two arrivals.

(c) The median time between the two successive arrivals.

(d) The probability that after one arrival it takes at least half an hour before the next arrival?

a

Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2

6. Suppose that the time spent online to do homework by a randomly selected student has a gamma

distribution with mean 20 minutes and variance 80 minutes2.

(a) What are the values of α and β?

(b) What is the probability that a student spends online at most 24 minutes?

(c) What is the probability that a student spends between 20 and 40 minutes online?

7. Suppose 𝑍~𝑁(0,1), find:

(a) 𝑃(𝑍 < 0.5)
(b) 𝑃(𝑍 = 0.5)
(c) 𝑃(𝑍 ≥ 2.3)
(d) 𝑃(−1.4 ≤ 𝑍 ≤ 0.6)
(e) The value of z0 such that 𝑃(|𝑍| ≤ 𝑧0 ) = 0.32
a
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
8. Suppose 𝑋~𝑁(5,1), find:
(a) 𝑃(𝑋 < 2.5)
(b) 𝑃(𝑋 ≥ 4.6)
(c) 𝑃(|𝑋| ≥ 3)
(d) 𝑃(|𝑋 − 5| ≥ 3)
9. Suppose that 𝑋~𝑁(µ, 𝜎 2 ) and that 𝑃(𝑋 ≤ 5) = 0.8413 and 𝑃(𝑋 ≥ 3.6) = 0.6554, what are the
values of µ and σ?
a
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
10. Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and
standard deviation 5 (“Mathematical Model of Chloride Concentration in Human Blood,” J. of Med.
Engr. and Tech., 2006: 25–30”).
(a) What is the probability that chloride concentration equals 104?
(b) What is the probability that chloride concentration is less than 104?
(c) What is the probability that chloride concentration is at most 104?
(d) What is the probability that chloride concentration differs from the mean by more than 1 standard
deviation? Does this probability depend on the values of µ or σ?
(e) What is the probability that someone’s blood chloride concentration level is between 99 and 114?
(f) What is the probability that someone’s blood chloride concentration level is above 120?
a
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
11. a Two safety inspectors inspect a new building and assign it a “safety score” of 1,2,3, or 4. Suppose
that the random variable X is the score assigned by the first inspector and the random variable Y is the
score assigned by the second inspector, and they have a joint probability mass function given below.
X
Y
1
2
3
4
1
2
3
4
0.09
0.02
0.01
0.00
0.03
0.15
0.01
0.01
0.01
0.03
0.24
0.02
0.01
0.01
0.04
0.32
(a) What is the probability that both inspectors assign the same safety score?
(b) What is the probability that the second inspector assigns a higher safety score than the first
inspector?
(c) What are the marginal probability mass function, expectation, and variance of the score assigned
by the first inspector?
(d) What are the marginal probability mass function, expectation, and variance of the score assigned
by the second inspector?
(e) Are the scores assigned by the two inspectors independent of each other?
(f) What are the covariance of the scores assigned by the two inspectors?
a
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2

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