Let X Neg Bin r p Use formulas of expected value and varianc

Let X~ Neg. Bin (r, p). Use formulas of expected value and variance and the equation x(x - 1 r - 1) = r(x r) to find E(X) and Var(X).

Solution

We have given that X~ Nag.Bin(r,p).

We have to use formuals of expected value and variance and the equation

x(x-1 C r-1) = r(x C r)

The p.m.f. of X is

P(Y=y) = [ (y+r-1) C y ] * (1-p)^y * p^r

We have to find E(X) and Var(X).

E(Y) = y *  [ (y+r-1) C y ] * (1-p)^y * p^r

   ⁼ (y+r-1) / (y-1)! (r-1)!* (1-p)^y * p^r

   ⁼ [r(1-p)] / p (y+r-1 C y-1) p^r+1 (1-p)^y-1

= r(1-p) /p (r+z C z) p^r+1 (1-p)^z

= r(1-p) / p

Let W1 denote the number of failures before the 1st success

W2 = number of failures between the 1st and 2nd success . . . . . .

Wr = number of failures between the (r 1)-th and r-th success

Since X is the total number of failures

X = W1 + W2 + . . . + Wr

Wi\'s are geometric random variables.

W1, W2, . . . Wr arise out of independent experiments

The variance of Wi is

Var(Wi) = q/p²

Var(X) = r*q / p²