GATE 2017 CS (CS2) – Question 41
For a discrete X with P(X=j)=pⱼ, pⱼ≥0, j=0,…,N and Σpⱼ=1, define $g_X(z)=\sum_{j=0}^Np_jz^j$. A variable Y has $g_Y(z)=(1-\beta+\beta z)^N$, 0≤β≤1. Find E[Y].
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Correct answer: (B) $N\beta$
Explanation
Differentiating a probability-generating function at one gives expectation: $g_Y\prime(1)=E[Y]$. Here the derivative is $N\beta(1-\beta+\beta z)^{N-1}$, which at z=1 equals Nβ. Thus B; Nβ(1−β) is the variance.