Suppose the probability mass function of a random variable X isgiven 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 NegativeBinomial 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 2August 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 thispart. (Hint: See Section 4.8.2 in Ross.) (d) In tossing a fair dierepeatedly (and independently on successive tosses), find theproba- bility of getting the third “1” on the xth toss. (Hint: LetX denote the number of tosses required until we get our third “1”,or equivalently our third success. Then X is what kind of randomvariable?) (e) In tossing a fair die repeatedly (and independentlyon successive tosses), find the proba- bility of getting the third“1” on the fifth toss. (f) What is the average number of trials itwill take to get our third “1”? (Hint: Use the results of part (c)for your solution).
