Math 3339
Homework 4 (Chapter 5)
Name:__________________________________ PeopleSoft ID:_______________
Instructions:
• Homework will NOT be accepted through email or in person. Homework must be submitted through
CANVAS BEFORE the deadline.
• Total possible points: 100
• You can use RStudio for any of these problems unless otherwise indicated.
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) Find 𝑃(0.4 < 𝑋 < 1).
(c) Find 𝑃(𝑋 < 0.4|𝑋 < 0.8).
(d) What is the mean value of this random variable?
(e) What is the variance of this random variable?
(f) Construct the cumulative distribution for this random variable.
a
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
, a 1
2. A random variable X takes values between 4 and 6 with a probability density function 𝑓(𝑥) = 𝑥𝑙𝑛(1.5)
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
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
Homework 4 (Chapter 5)
Name:__________________________________ PeopleSoft ID:_______________
Instructions:
• Homework will NOT be accepted through email or in person. Homework must be submitted through
CANVAS BEFORE the deadline.
• Total possible points: 100
• You can use RStudio for any of these problems unless otherwise indicated.
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) Find 𝑃(0.4 < 𝑋 < 1).
(c) Find 𝑃(𝑋 < 0.4|𝑋 < 0.8).
(d) What is the mean value of this random variable?
(e) What is the variance of this random variable?
(f) Construct the cumulative distribution for this random variable.
a
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
, a 1
2. A random variable X takes values between 4 and 6 with a probability density function 𝑓(𝑥) = 𝑥𝑙𝑛(1.5)
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
Probability And Statistics for Engineers And Scientists, Hayter, A. (2007), Chapter 2
