Find the m g f of the pdf 𝑓(𝑥) = 𝑞𝑥−1𝑝, 𝑥 = 1,2,3, . . .. and hence evaluate its mean andvariance. (given p+q =1)

The moment generating function (m.g.f.) of a random variable X is defined as M(t) = E(e^(tX)).

Given the probability mass function (p.m.f) f(x) = qx^(p-1), x = 1,2,3,..., we can find the m.g.f. by summing over all possible values of x, weighted by their probabilities.

M(t) = Σ [e^(tx) * f(x)] = Σ [e^(tx) * qx^(p-1)] = Σ [q * e^(tx) * x^(p-1)] = q * Σ [x^(p-1) * e^(tx)]

This is a geometric series with common ratio e^t. The sum of a geometric series a + ar + ar^2 + ... is given by a / (1 - r), where a is the first term and r is the common ratio.

In this case, a = e^t and r = e^t, so the sum is

M(t) = q * e^t / (1 - e^t)

The mean (expected value) of X is given by E(X) = M'(0), the derivative of the m.g.f. evaluated at t = 0.

M'(t) = q * e^t / (1 - e^t)^2

E(X) = M'(0) = q / (1 - 1)^2 = q

The variance of X is given by Var(X) = M''(0) - [M'(0)]^2, the second derivative of the m.g.f. evaluated at t = 0 minus the square of the first derivative evaluated at t = 0.

M''(t) = 2q * e^(2t) / (1 - e^t)^3

Var(X) = M''(0) - [M'(0)]^2 = 2q - q^2

So, the mean and variance of X are q and 2q - q^2, respectively.