Question

Suppose the probability mass function of a random variable X is given by

??x−1?pr(1−p)x−r, ifx=r,r+1,r+2,... f(x) = r−1

0, otherwise

If this is the case then we say X is distributed as a Negative Binomial Random Variable with parameters r and p and we write X ∼ NegBin(r, p) (a) If we set r = 1, what distribution do we get? (b) Explain what this random variable models and justify the formula. (Hint: See Section 4.8.2 in Ross.) Math 241 Quiz 3 - Page 2 of 2 August 2019 (c) For this random variable, what is E[X] and Var[X]? There is no need to prove your answer or show any work for this part. (Hint: See Section 4.8.2 in Ross.) (d) In tossing a fair die repeatedly (and independently on successive tosses), find the proba- bility of getting the third “1” on the xth toss. (Hint: Let X denote the number of tosses required until we get our third “1”, or equivalently our third success. Then X is what kind of random variable?) (e) In tossing a fair die repeatedly (and independently on successive tosses), find the proba- bility of getting the third “1” on the fifth toss. (f) What is the average number of trials it will take to get our third “1”? (Hint: Use the results of part (c) for your solution).

Answer 1

The PMF of Negative Binomial random variable is

This is the probability of getting the r-the success in the x-the toss.

a) When , the PMF is

This is a Geometric distribution with parameter .

b)This is the probability of getting the the first success in the x-the toss. The success probability in any toss is independent of any other tosses.

c) The expected value and variance of the Negative Binomial random variable are

d) Here . Then,

X is a Negative Binomial random variable.

e) Here , then

f) We need to find the expected value of the Negative Binomial distribution.