Random Variables and Probability Distributions

The probability distribution of a discrete random variable $X$ is a listing of each possible value $x$ taken by $X$ along with the probability $P(x)$ that $X$ takes that value in one trial of the experiment. 

The mean $\mu$ of a discrete random variable $X$ is a number that indicates the average value of $X$ over numerous trials of the experiment. It is computed using the formula $\mu=\Sigma x P(x)$. 

The variance $\sigma^{2}$ and standard deviation $\sigma$ of a discrete random variable $X$ are numbers that indicate the variability of $X$ over numerous trials of the experiment. They may be computed using the formula $\sigma^{2}=\left[\Sigma x^{2} P(x)\right]-\mu^{2}$, taking the square root to obtain $\sigma$.