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7.36 If X, and X, are (Hint: Use Theorem 1.12.). independent random variables having the geometric dis- tribution with the parameter 8, show that Y=X, + X, is a random variable having the negative binomial distribution with the parameters and k = 2. 7.37 II X and Y are independent random variables having the standard normal distribution, show that the random variable Z = X + Y is also normally distributed. (Hint: Complete the square in the exponent.) What are the mean and the variance of this normal distribution?

7.36 If X, and X, are (Hint: Use Theorem 1.12.). independent random variables having the geometric dis- tribution with the parameter 8, show that Y=X, + X, is a random variable having the negative binomial distribution with the parameters and k = 2. 7.37 II X and Y are independent random variables having the standard normal distribution, show that the random variable Z = X + Y is also normally distributed. (Hint: Complete the square in the exponent.) What are the mean and the variance of this normal distribution?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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(Hint: Use Theorem 1.12.) 7.36 If X, and X, are independent random variables having the geometric dis- tribution with the parameter 8, show that Y=X, + X₂ is a random variable having the negative binomial distribution with the parameters and k = 2. 7.37 IX and Y are independent random variables having the standard normal distribution, show that the random variable Z = X + Y is also normally distributed. (Hint: Complete the square in the exponent:) What are the mean and the variance of this normal distribution?
Transcribed Image Text:(Hint: Use Theorem 1.12.) 7.36 If X, and X, are independent random variables having the geometric dis- tribution with the parameter 8, show that Y=X, + X₂ is a random variable having the negative binomial distribution with the parameters and k = 2. 7.37 IX and Y are independent random variables having the standard normal distribution, show that the random variable Z = X + Y is also normally distributed. (Hint: Complete the square in the exponent:) What are the mean and the variance of this normal distribution?
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