GATE 2020 ST – Question 56

Joint Distributions · Joint and Conditional Distributions · 2 marks · Numerical answer

Joint mass is $P(X=x,Y=y)=\binom{x}{y}(1/4)^x$ for x=1,2,… and y=0,…,x, zero otherwise. Find Var(Y).

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Correct answer: 1 to 1

Explanation

Summing over y gives P(X=x)=2ˣ/4ˣ=2⁻ˣ, so X is geometric on1,2,… with mean2 and variance2. Given X, Y is Binomial(X,1/2). Total variance is E[X/4]+Var(X/2)=2/4+2/4=1.

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