GATE 2025 ST – Question 41

Joint Distributions · Joint and Conditional Distributions · 2 marks · Multiple choice

For λ>0, joint masses are p_(X,Y)(m,n)=λⁿe^(−λ)/[2ⁿm!(n−m)!],m=0,…,n,n=0,1,…. Which holds?

X is Poisson with mean λ.
Y is Poisson with mean 2λ.
X|Y=3 is Binomial(3,1/2).
E(Y|X=2)=λ/2

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Correct answer: (C) X|Y=3 is Binomial(3,1/2).

Explanation

Factor the mass as [e^(−λ)λⁿ/n!]·[C(n,m)2⁻ⁿ]. Thus Y~Poisson(λ) and X|Y=n~Bin(n,1/2), proving C. Poisson thinning gives X~Poisson(λ/2) and Y−X independent Poisson(λ/2), so E(Y|X=2)=2+λ/2. **C**.

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