A random variable X has the discrete uniform distribution...

A random variable $X$ has the discrete uniform distribution
$$
f(x)=\frac{1}{m}, \quad x=1,2, \ldots, m
$$
(a) Show that the moment-generating function is $M_{X}(t)=\frac{e^{t}\left(1-e^{t m}\right)}{m\left(1-e^{t}\right)}$
(b) Use $M_{X}(t)$ to find the mean and variance of $X$.

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Key Concepts

Moment Generating Function

The moment generating function (MGF) of a random variable is defined as M_X(t) = E(e^(tX)). It is a tool used in probability theory to encapsulate all the moments of a random variable. By differentiating the MGF with respect to t and evaluating at t = 0, one can obtain the mean, variance, and higher moments of the distribution.

Discrete Uniform Distribution

A discrete uniform distribution is one in which a finite number of outcomes are equally likely. Each outcome has the same probability, and the distribution is typically defined over a set of consecutive integers. This type of distribution is fundamental in probability theory as it represents a situation with complete symmetry among possible outcomes.

Using the MGF to Derive Moments

By differentiating the moment generating function with respect to t, one can find the moments of the distribution. Specifically, the first derivative evaluated at t = 0 gives the mean (expected value) of the random variable, and the second derivative (along with the first) can be used to calculate the variance. This method provides a systematic way of obtaining the moments and understanding the variability of the distribution.

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