What is the formula for the variance of a probability distribution?

The formula for the variance of a probability distribution depends on whether you are working with a discrete or continuous distribution.

For a discrete probability distribution, the variance is calculated using the following formula:

Var(X) = Σ[(x - μ)² * P(x)]

Where:

Var(X) represents the variance of the random variable X.
Σ denotes the summation symbol.
x represents each possible value of X.
μ represents the mean (expected value) of X.
P(x) represents the probability mass function of X, which gives the probability of each value x.
For a continuous probability distribution, the variance is calculated using integration. The formula is as follows:

Var(X) = ∫[(x - μ)² * f(x)] dx

Where:

Var(X) represents the variance of the random variable X.
∫ represents the integration symbol.
x represents the variable of integration.
μ represents the mean (expected value) of X.
f(x) represents the probability density function (PDF) of X, which gives the probability density at each point x.
It's important to note that these formulas assume that the random variable X has a well-defined mean and variance.