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Mean, Variance, and Expected Values

These are fundamental concepts in statistics that describe the central tendency and variability of a probability distribution.

Mean​

The mean, μ\mu, of a probability provides a measure of the central tendency. For a discrete random variable XX with a probability mass function P(x)\mathbb{P}(x):

μ=∑Xx⋅P(x)\mu = \sum_X x \cdot \mathbb{P}(x)

For a continuous random variable XX with a probability density function f(x)f(x):

μ=∫−∞∞x⋅f(x) dx\mu = \int_{-\infty}^{\infty} x \cdot f(x) \, dx

The mean can also be expressed in term of the expected value, i.e. the weighted average value of all possible outcomes of the random variable.

μ=E(X)\mu = E(X)

Variance​

Variance measures the spread or dispersion of a set of values. It is the expected value of the squared deviation of a random variable from its mean.

Var(X)=σ2=E[(X−μ)2]\text{Var}(X) = \sigma^2 = E[(X - \mu)^2] σ2={∫−∞∞(x−μ)2f(x) dxx continuous∑X(x−μ)2P(x)x discrete\sigma^2 = \begin{cases} \int_{-\infty}^\infty(x-\mu)^2f(x)\,dx & \text{$x$ continuous}\\ \sum_X (x-\mu)^2\mathbb{P}(x) & \text{$x$ discrete} \end{cases}

Standard Deviation​

The standard deviation is the square root of the variance and provides a measure of the dispersion in the same units as the mean.

σ=Var(X)\sigma = \sqrt{\text{Var}(X)}