Let X be a discrete random variable, then the variance of X is :

Answer: The variance of a discrete random variable $X$ is given by the formula $\text{Var}(X) = E[X^2] - (E[X])^2$ .

Step-by-step solution

Step 1: Define Variance

The variance of a discrete random variable $X$ , denoted as $\text{Var}(X)$ , is defined as the expected value of the squared difference between the random variable and its mean. This measures how far the values of the random variable are spread out from the mean.

Step 2: Expand the Expectation

We can expand the squared term inside the expectation. Let $E[X] = \mu$ . Then $(X - \mu)^2 = X^2 - 2X\mu + \mu^2$ .

Step 3: Apply Linearity of Expectation

Using the linearity property of expectation, $E[aY + bZ] = aE[Y] + bE[Z]$ , and $E[c] = c$ for a constant $c$ . Here, $E[X]$ is a constant, so $E[2XE[X]] = 2E[X]E[X]$ and $E[(E[X])^2] = (E[X])^2$ .

Step 4: Simplify the Expression

Combining the terms, we get the commonly used formula for the variance of a discrete random variable: the expected value of $X^2$ minus the square of the expected value of $X$ .

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