The continuous uniform random variable X has density...

The continuous uniform random variable $X$ has density function $$ F(x)=\frac{1}{\beta-\alpha}, \quad \alpha \leq x \leq \beta $$ (a) Show that the moment-generating function is $$ M_{X}(t)=\frac{e^{t \beta}-e^{t \alpha}}{t(\beta-\alpha)} $$ (b) Use $M_{X}(t)$ to find the mean and variance of $X$.

The continuous uniform random variable $X$ has density function
$$
F(x)=\frac{1}{\beta-\alpha}, \quad \alpha \leq x \leq \beta
$$
(a) Show that the moment-generating function is
$$
M_{X}(t)=\frac{e^{t \beta}-e^{t \alpha}}{t(\beta-\alpha)}
$$
(b) Use $M_{X}(t)$ to find the mean and variance of $X$.

Key Concepts

Differentiation of the MGF for Moment Calculation

To calculate the mean and variance using the MGF, one differentiates the MGF with respect to t and evaluates the derivative(s) at t = 0. The first derivative gives the mean (the first moment), and the second derivative, combined with the first, is used to compute the variance. This approach leverages the powerful property that the MGF, if it exists, uniquely determines the moments of the distribution.

Continuous Uniform Distribution

This is a probability distribution where all outcomes in a specific interval [?, ?] are equally likely. The density function is constant over the interval, with its value being 1/(???). It is a foundational concept in probability theory for modeling experiments with equal likelihood across a range of outcomes.

Density Function

The density function of a continuous random variable characterizes the probability distribution by assigning probabilities to intervals via integration. For the uniform distribution, the density is constant over the interval and zero elsewhere, which simplifies the computation of probabilities, expectations, and other moments.

Moment-Generating Function (MGF)

The MGF of a random variable, defined as M_X(t) = E(e^(tX)), provides a compact way to encapsulate all the moments of the distribution. Differentiating the MGF with respect to t and evaluating at 0 yields the moments (mean, variance, etc.) of the variable. It is an essential tool in probability theory for analyzing distributions.

Expectation and Variance

Expectation is the first moment, representing the average or mean value of a random variable, while variance measures the spread or dispersion around the mean. In the context of the uniform distribution, these moments can be computed directly from the integral definitions or derived via the moment-generating function, providing insights into the distribution's central tendency and variability.

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