Final answer:
To calculate the MGF for the random variable X with the given PMF, we use the definition of MGF and ensure the convergence of the series. Using the MGF, the mean and variance of X can be found by taking the first and second derivatives of MGF with respect to t, evaluated at t=0.
Step-by-step explanation:
Calculating the Moment Generating Function (MGF)
To compute the moment generating function MX(t) for the random variable X, we can use the definition of the MGF and the given probability mass function (PMF). The MGF of a random variable X is defined as MX(t) = E(etX), where E denotes the expected value.
The PMF given is as follows:
• P(X=0) = 2/5
• P(X=k) = ((3/4)k)(1/5), for k=1,2,...
The moment generating function MX(t) is then:
MX(t) = P(X=0)e0t + ∑k=1∞ P(X=k)ekt
Plugging the PMF into the MGF formula gives:
MX(t) = (2/5) + ∑k=1∞ ((3/4)k)(1/5)ekt
To ensure the MGF is not infinite, we must check the convergence of the series. This series converges when |3/4 * et| < 1.
Finding Mean and Variance Using MGF
To find the mean and variance using the MGF:
1. The mean of X, or E(X), can be computed by taking the first derivative of MX(t) with respect to t and evaluating it at t = 0.
2. To find the variance Var(X), we take the second derivative of MX(t) with respect to t and evaluate at t = 0, and then subtract the square of the mean obtained in step 1.