Probability - Mean and Variance of a Random Variable

Discrete Random Variable: A random variable $X$ is a real-valued function whose domain is the sample space of a random experiment. For a discrete variable, the values can be listed as a sequence $x_1, x_2, ...$ which, when plotted on a horizontal axis, appear as distinct, separated points.

Probability Distribution Function (PDF): A mapping that assigns a probability $p_i$ to each possible value $x_i$. Visually, this is represented by a probability histogram where each outcome $x_i$ has a bar of height $p_i$, and the total area or sum of the heights of all bars must equal $1$.

Mean or Expected Value $E(X)$: This is the weighted average of all possible values of $X$. In a physical sense, if the probabilities were masses placed at positions $x_i$ on a beam, the mean would be the 'center of gravity' or the exact point where the beam would balance.

Variance $Var(X)$: A statistical measure that describes the dispersion of the random variable values around the mean. On a graph, a 'flat' or 'wide' distribution indicates a high variance, while a 'tall' and 'narrow' distribution indicates the values are clustered closely around the mean.

Standard Deviation $\sigma$: This is the positive square root of the variance. It is often more intuitive than variance because it is expressed in the same units as the random variable $X$, representing the typical distance an outcome is from the mean.

Linearity of Expectation: A concept stating that for constants $a$ and $b$, $E(aX + b) = aE(X) + b$. Visually, adding $b$ shifts the entire probability distribution graph to the right, while multiplying by $a$ stretches or compresses the distribution.

Variance of Linear Transformations: The variance of a transformed variable is given by $Var(aX + b) = a^2 Var(X)$. This highlights that shifting a distribution (adding $b$) does not change its spread, but scaling it (multiplying by $a$) increases the spread by the square of the scale factor.