Question

Jim and Joe draw one at a time, so this is “with replacement"

An urn contains 5 marbles, numbered 1 to 5. Jim and Joe each drew a marble, recorded the number then returned it to the urn. Let X be the difference between the larger and smaller number. (e.g. if one got 4 and the other 1 then X=3).

Find the probability that X =0

Find the probability that X =3

Find the Expectation of X:

Find the Expectation of E[ X^2]

Find Var(X)

Answer 1

1)P(X=0) =P(both draw the same number) =5/25 =1/5 (since 5 such outcome where both number are same)

2)P(X=3) =P(Jim 1 and Joe 4)+P(Jim 2 and Joe 5)+P(Jim 4 and Joe 1)+P(Jim 5 and Joe 2) =1/25+1/25+1/25+1/25 =4/25

3) below is pmf of X:

P(X=0) =1/5

P(X=1) =8/25

P(X=2) =6/25

P(X=3) =4/25

P(X=4) =2/25

E(X)=xP(x)=0*1/5+1*8/25+2*6/25+3*4/25+4*2/25=1.6

E(X2)=x2P(x)=0^2*1/5+1^2*8/25+2*6/25+3^2*4/25+4^2*2/25=4

Var(X)=E(X2)-(E(X))^2 -4-1.6^2 =1.44