Suppose a discrete uniform random variable X assumes only...

Suppose a discrete uniform random variable $X$ assumes only the values $C+L, C+2 L, \cdots, C+n L$, where $C, n$, and $L$ are constants. Show that $$ E[X]=C+\frac{(n+1)}{2} L, E\left[X^2\right]=C^2+(n+1) L C+\frac{(n+1)(2 n+1)}{6} L^2, $$ and $$ \operatorname{Var}[X]=\frac{n^2-1}{12} L^2 $$

Suppose a discrete uniform random variable $X$ assumes only the values $C+L, C+2 L, \cdots, C+n L$, where $C, n$, and $L$ are constants. Show that
$$
E[X]=C+\frac{(n+1)}{2} L, E\left[X^2\right]=C^2+(n+1) L C+\frac{(n+1)(2 n+1)}{6} L^2,
$$
and
$$
\operatorname{Var}[X]=\frac{n^2-1}{12} L^2
$$

Key Concepts

Discrete Uniform Distribution

A discrete uniform distribution is a probability distribution where a random variable can take on a finite number of distinct values, each with an equal likelihood. This uniformity simplifies the process of calculating statistical measures like the mean and variance because every outcome contributes equally to these computations.

Expectation

Expectation, or the mean of a random variable, is the weighted average of all possible values the variable can assume, where each value is weighted by its corresponding probability. In the context of a discrete uniform distribution, this simplifies to the arithmetic average of the possible outcomes because each outcome has the same probability of occurring.

Variance

Variance measures the spread or dispersion of a random variable's values around its mean. It is computed by taking the expected value of the squared differences between the random variable's values and the mean. In a uniform distribution, the symmetry and equal likelihood of outcomes simplify the derivation of a closed-form expression for the variance.

Summation Formulas

Summation formulas for arithmetic sequences, such as the formula for the sum of the first n integers and the sum of squares of the first n integers, are fundamental tools in deriving exact expressions for statistical measures like the expectation and variance when dealing with uniformly spaced outcomes.

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