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.
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)=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
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