Let the moment-generating function of a random variable X be...

Let the moment-generating function of a random variable $X$ be given by
$$
M_{X}(t)= \begin{cases}\frac{1}{t}\left(e^{t / 2}-e^{-t / 2}\right) & \text { if } t \neq 0 \\ 1 & \text { if } t=0 .\end{cases}
$$
Find the distribution function of $X$.

Key Concepts

Moment-Generating Function (MGF)

The moment-generating function is a tool in probability theory defined as M(t) = E[e^(tX)], which, when it exists in an open interval around t = 0, uniquely determines the probability distribution of a random variable. It is used to compute moments by differentiation and to identify distributions by matching the form of a given MGF with known MGFs.

Uniform Distribution

A uniform distribution is characterized by a constant probability density function over a specified interval, meaning every outcome in that interval is equally likely. When the MGF of a random variable matches the form associated with a uniform distribution, one can deduce that the variable is uniformly distributed and then derive its density and cumulative distribution function accordingly.

Inversion of the Moment-Generating Function

Inversion of an MGF involves matching the provided moment-generating function with known MGFs to determine the corresponding probability distribution. This process requires familiarity with standard forms and properties of MGFs so that one can identify the distribution from its unique moment structure.

Cumulative Distribution Function (CDF)

The cumulative distribution function of a random variable provides the probability that the variable takes on a value less than or equal to a given number. Once the probability density function is derived from the identified distribution, the CDF can be computed by integrating the density function over the appropriate interval, thereby fully characterizing the distribution.

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