Question

From textbook:

"Let X have a negative binomial distribution with parameters r and p such that:

p(k) = p' (1-p)*-, k=r,r+1,..

Find E[X] and Var[X] without using the definition; instead, consider how X can be written as a sum of independent random variables."

Question: How do I do this?

Answer 1

Here the pmf for ! Negative Binomial distribution is is f**)=P(K) = ) pr (1-P) KY; Kr, rt, . on Note that the negative Binomial dith is Sometimes defined in terms of the random de variable y Niemser of failures before rith suceers This formulation is statistically equivalent to the one gire abrue in terms of x= ctrias at which the rth oncees occurs Since . Y=x-r . Then the alternate form of neg. Bin. dith is P(Y= y) = fragil PR (1-P) *; *=0,13 --

Turm Tum Ely)= Ë ( 113.") p* (-2) ? Ž, (9*") ** (propos Typ Ž (721) pol (ap) ? 2-6 = (1-P) P XI or/1-P) P Note that the bracket post is is Negative Binomial mars. furetion which est satiefy I fcm) ; So, without Calculation we can say as the above Similar expansion that V(Y) = (1-0 P2 Now, x oan be written as ... . x= Geometric Random 9, +42*-. 't ...yo variasle, where in the sum of y; in Geolp).