Question

Calculate the moment generating function of a geometric random variable.

Short answer

The moment generating function (MGF) of a geometric random variable X, with probability p of success, is given by: M_X(t) = \(p*\frac{e^t}{1 - (1-p)e^t}\).

Step-by-step solution

Define the probability mass function for a geometric random variable

The probability mass function for a geometric random variable X is given by: P(X=k) = (1-p)^(k-1)p, where k is the number of trials and p is the probability of success on each trial.

Define the moment generating function for a random variable

The moment generating function for a random variable X is given by the formula: M_X(t) = E(e^(tX)), where E denotes the expected value, and t is a real number.

Calculate the expected value of e^(tX) using the given probability mass function

To calculate the expected value of e^(tX), we will use the formula: E(e^(tX)) = Σ [P(X=k) * e^(tk)], where the summation is over all positive integers k. Now, we have: E(e^(tX)) = Σ [(1-p)^(k-1)p * e^(tk)].

Factor out the common terms from the summation

We can factor out the p from the summation: E(e^(tX)) = p*(Σ ((1-p)^(k-1) * e^(tk)). Now we need to find a closed-form expression for the summation.

Simplify the summation and calculate the MGF

Notice that the summation is a geometric series: Σ ((1-p)^(k-1) * e^(tk)), with common ratio (1-p)*e^t. So we have: Σ ((1-p)^(k-1) * e^(tk))= (e^t) / (1 - (1-p)e^t), provided that the denominator is nonzero. Therefore, the MGF for a geometric random variable will be: M_X(t) = p*(e^t) / (1 - (1-p)e^t). This is the moment generating function for a geometric random variable.