Show that for a discrete uniform random variable X, if each...

Show that for a discrete uniform random variable $X$, if each of the values in the range of $X$ is multiplied by the constant $c,$ the effect is to multiply the mean of $X$ by $c$ and the variance of $X$ by $c^{2}$. That is, show that $E(c X)=c E(X)$ and $V(c X)=c^{2} V(X).$

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Key Concepts

Variance Scaling Property

The variance scaling property describes how the variance of a random variable changes under a linear transformation. When a random variable X is multiplied by a constant c, its variance is multiplied by the square of that constant, i.e., Var(cX) equals c² times Var(X), reflecting the spread of the distribution under scaling.

Discrete Uniform Random Variable

A discrete uniform random variable is a type of random variable characterized by a finite set of outcomes, each of which has an equal probability of occurring. This concept is central in probability theory and statistical inference when each outcome in the sample space is considered equally likely.

Linearity of Expectation

The linearity of expectation is a fundamental property in probability theory that states the expected value of a linear transformation of a random variable equals the transformation applied to the expected value. Specifically, for any constant c and random variable X, the expectation of cX is c times the expectation of X, a property that holds regardless of the distribution of X.

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