2.7 Key definitions and Relationships

A random variable (rv) is a map from $\mathrm{\Omega }\to ℝ$.

The cumulative distribution function (cdf) of any rv, $X$, is $F_X(x)=𝖯\left(X\le x\right)$; $x$ can be any real number.

Discrete random variables can take at most a countably infinite number of values. The probability mass function (pmf) of a discrete rv is $p_X(x)=𝖯\left(X=x\right)$.

Continuous random values can take a continuum of values. The probability density function (pdf) of a continuous rv is $f_X(x)=\frac{\mathrm{d}}{\mathrm{d}x}{F}_X(x)$. Conversely, $F_X(x)={\int }_{-\mathrm{\infty }}^x{f}_X(t)dt$.

The quantile function of a continuous rv, $X$, evaluated at some $p\in (0,1]$ is the smallest value $x$ such that $F_X(x)=p$.

For some real-valued function, $g$, the expectation, $𝖤\left[g(X)\right]$ is ${\sum }_{i=-\mathrm{\infty }}^{\mathrm{\infty }}{p}_X(i)g(i)$ if $X$ is discrete and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_X(t)g(t)dt$ if $X$ is continuous.

Expectation is linear: $𝖤\left[ag(X)+bh(X)\right]=a𝖤\left[h(X)\right]+b𝖤\left[g(X)\right]$.

The variance of any rv, $X$, is $\mathrm{𝖵𝖺𝗋}\left[X\right]=𝖤\left[{(X-𝖤\left[X\right])}^2\right]=𝖤\left[X^2\right]-𝖤{\left[X\right]}^2$. The standard deviation is $\mathrm{𝖲𝗍𝖽𝖣𝖾𝗏}\left[X\right]=\sqrt{\mathrm{𝖵𝖺𝗋}\left[X\right]}$.

$\mathrm{𝖵𝖺𝗋}\left[aX+b\right]=a^2\mathrm{𝖵𝖺𝗋}\left[X\right]$.