7.36 If X, and X, are 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.

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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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