GATE CS 2017 Set 2 Q31 — Probability: Random Variables and Distributions

The given polynomial function $g_X(z) = \sum_{j=0}^N p_j z^j$ is the Probability Generating Function (PGF) of the discrete random variable $X$.

For the random variable $Y$, the PGF is given as: $$g_Y(z) = (1 - \beta + \beta z)^N$$

Method 1: Using the derivative of the PGF

The expectation (mean) of a discrete random variable $Y$ can be computed by evaluating the first derivative of its PGF at $z = 1$: $$E[Y] = g'_Y(1)$$

Differentiating $g_Y(z)$ with respect to $z$: $$g'_Y(z) = \frac{d}{dz} \left[ (1 - \beta + \beta z)^N \right]$$ $$g'_Y(z) = N(1 - \beta + \beta z)^{N-1} \cdot \beta$$

Evaluating this derivative at $z = 1$: $$E[Y] = g'_Y(1) = N(1 - \beta + \beta(1))^{N-1} \cdot \beta$$ $$E[Y] = N(1)^{N-1} \cdot \beta = N\beta$$

Method 2: Recognizing the distribution

The PGF of a Binomial random variable with parameters $N$ (number of trials) and $p$ (probability of success) is: $$G(z) = (q + pz)^N = (1 - p + pz)^N$$

Comparing this with $g_Y(z) = (1 - \beta + \beta z)^N$, we can see that $Y$ is a Binomial random variable with parameters $N$ and $p = \beta$.

The expectation of a Binomial random variable is: $$E[Y] = Np = N\beta$$

Thus, the correct option is B.