Question

Roll a fair die and denote the outcome by Y . Then flip Y many fair coins and let X denote the number of tails observed. Find the probability mass function and expectation of X.

Answer 1

below is probability mass function of X:

P(X=0)=P(Y=1)*P(X=0|Y=1)+P(Y=2)*P(X=0|Y=2)+P(Y=3)*P(X=0|Y=3)+P(Y=4)*P(X=0|Y=4)+P(Y=5)*P(X=0|Y=5)+P(Y=6)*P(X=0|Y=6)=(1/6)*(¹C₀(1/2)⁰(1/2)¹)+²C₀(1/2)⁰(1/2)²)+³C₀(1/2)⁰(1/2)³)+⁴C₀(1/2)⁰(1/2)⁴)+⁵C₀(1/2)⁰(1/2)⁵)+⁶C₀(1/2)⁰(1/2)^6\))

=(1/6)*(0.984375)=0.1640625

P(X=1)=P(Y=1)*P(X=1|Y=1)+P(Y=2)*P(X=1|Y=2)+P(Y=3)*P(X=1|Y=3)+P(Y=4)*P(X=1|Y=4)+P(Y=5)*P(X=1|Y=5)+P(Y=6)*P(X=1|Y=6)=0.3125

P(X=2)=0.2578125

P(X=3)=0.1666667

P(X=4)=0.075521

P(X=5)=0.020833

P(X=6)=0.002604

hence expectation of X: E(X)=$\sum$xP(x)

=0*0.1640625+1*0.3125+2*0.2578125+3*0.166667+4*0.075521+5*0.020833+6*0.002604=1.75