Consider a box containing 3 red ball and 7 white balls Suppo

Consider a box containing 3 red ball and 7 white balls. Suppose that balls are drawn one at a time, at random, withought repplacment from this box until two red balls are obtained. Let X deonte the number of draws on which the second red ball is obtained.

a) find the p.m.f of X

b) find the expected value of X

c) find the expected value of X^2 ( I belive I can do this one)

d) Find variance of X

Solution

a)

P(X=2)=3/10 * 2/9 = 0.07

P(X=3)=3/10 * 2/9 * 7/8 = 0.06

P(X=4)=3/10 * 2/9 * 7/8 * 6/7 = 0.05

P(X=5)=3/10 * 2/9 * 7/8 * 6/7 * 5/6 = 0.04

P(X=6)=3/10 * 2/9 * 7/8 * 6/7 * 5/6 * 4/5 = 0.03

P(X=7)=3/10 * 2/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 = 0.02

P(X=8)=3/10 * 2/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 * 2/3 = 0.02

P(X=9)=3/10 * 2/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 * 2/3 * 1/2 = 0.01

b)

E(X) = 2*0.07 + 3*0.06+4*0.05+5*0.04+6*0.03+7*0.02+8*0.02+9*0.01 = 1.29 Answer

c)

E(X^2) = 2^2*0.07 + 3^2*0.06+4^2*0.05+5^2*0.04+6^2*0.03+7^2*0.02+8^2*0.02+9^2*0.01 = 6.77 Answer

d)

Var(X) = E(X^2) - (E(X)^2

= 6.77 - (1.29)^2

= 5.11 Answer