Probability Distributions: PMFs and PDFs

Formula Name	PMFs (Discrete Random Variable)	PDFs (Continuous Random Variable)
Expectation	$E[X]=\sum _{x}x\cdot p_X(x)$	$E[X]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\cdot f_X(x)dx$
Description	The expectation (or mean) is the sum of all possible values of $X$ weighted by their probabilities. It represents the average value of $X$.	The expectation (or mean) is the integral of $x$ times the PDF over all possible values of $x$. It represents the average value of $X$.
Variance	$Var(X)=\sum _x(x-E[X]{)}^2\cdot p_X(x)$	$Var(X)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-E[X]{)}^2\cdot f_X(x)dx$
Description	The variance measures how much the values of $X$ deviate from the mean. It is the weighted average of the squared deviations from the mean.	The variance measures the spread of $X$ around its mean. It is the integral of the squared deviations from the mean, weighted by the PDF.
Standard Deviation	$SD(X)=\sqrt{Var(X)}$	$SD(X)=\sqrt{Var(X)}$
Description	The standard deviation is the square root of the variance. It provides a measure of spread in the same units as the random variable itself.	The standard deviation is the square root of the variance and measures the spread of $X$ around its mean in the same units as $X$.