Question

The number of home runs hit by a certain team in one game is a random variable with the following distribution:
X = 0, P(X = 0) = 0.4
X = 1, P(X = 1) = 0.4
X = 2, P(X = 2) = 0.2

The team plays 2 games. The number of home runs hit in one game, is independent of the number of home runs in the other game. Let Y be the total number of home runs. Find E(Y) and Var(Y).

Answer 1

here Y=x1+x2 ; where x1 and x2 are home run hit on 2 games

below is pmf of Y:

P(Y=0) =P(x1=0)*P(x2=0)=0.4*0.4=0.16

P(Y=1)=P(x1=0)P(x2=1)+P(x1=1)P(x2=0) =0.4*0.4+0.4*0.4 =0.32

P(Y=2)=P(X1=0)*P(x2=2)+P(X1=2)*P(x2=0)+P(X1=1)*P(x2=1)

=0.4*0.2+0.2*0.4+0.4*0.4=0.32

P(Y=3)=P(X1=1)P(X2=2)+P(X1=2)P(X2=1)=0.4*0.2+0.2*0.4 =0.16

P(Y=4)=P(X1=2)P(X2=2) =0.2*0.2 =0.04

from above

E(Y)=yP(y)=0*0.16+1*0.32+2*0.32+3*0.16+4*0.04 =1.6

E(Y2)=y2P(y)=0^2*0.16+1^2*0.32+2^2*0.32+3^2*0.16+4^2*0.04 =3.68

Var(y)=E(Y2)-E(Y)2 =3.68-1.6^2=1.12