Probability Density Functions and Expected Value

A probability density function (PDF) \(f\) of a single random variable \(X\) on a domain \(D\) is defined by the property that \(\int_{D} f(x) \,\mathrm{d}x = 1\,.\) For an interval \(I,\) the probability that \(X\) falls within that interval is denoted as \(\operatorname{P}(X \in I)\) and computed as \(\int_{I} f(x) \,\mathrm{d}x.\) The cumulative distribution function \(F\) of the PDF \(f\) is the antiderivative of \(f\) such that \(F(x) = \int_{-\infty}^{x} f(u)\,\mathrm{d}u.\) In particular, \(F(x) = \operatorname{P}(X \leq x),\) and for an interval \({I = [a,b]}\) we have \(\operatorname{P}(X \in I) = F(b) - F(a).\) The expected value of \(X\), denoted \(\operatorname{E}[X],\) also often called the mean and denoted \(\mu,\) is computed as \(\operatorname{E}[X] = \int_{-\infty}^{\infty} {\color{maroon} x} f(x) \,\mathrm{d}x\) — it’s the first moment of the PDF. The variance of \(X\), denoted \(\operatorname{Var}(X),\) taken by definition to be the square of the standard deviation and denoted \(\sigma^2,\) is computed as \(\operatorname{E}\bigl[(X\!-\!\mu)^2\bigr] = \int_{-\infty}^{\infty} {\color{maroon} (x\!-\!\mu)^2} f(x) \,\mathrm{d}x\) — it’s the second central moment of the PDF, its weight being measured as a distance from the mean \(\mu.\) Higher moments of a PDF are often computed as standardized central moments, their weights being measured as units of the standard deviation \(\sigma\) from the mean \(\mu.\)

\( \displaystyle \operatorname{E}\Biggl[\biggl(\frac{X\!-\!\mu}{\sigma}\biggr)^{\!3}\Biggr] =\!\!\!\! \int\limits_{-\infty}^{\infty} {\color{maroon} \biggl(\frac{x\!-\!\mu}{\sigma}\biggr)^{\!3}} f(x) \,\mathrm{d}x\)

skewness \(\gamma_1\)

\( \displaystyle \operatorname{E}\Biggl[\biggl(\frac{X\!-\!\mu}{\sigma}\biggr)^{\!4}\Biggr] =\!\!\!\! \int\limits_{-\infty}^{\infty} {\color{maroon} \biggl(\frac{x\!-\!\mu}{\sigma}\biggr)^{\!4}} f(x) \,\mathrm{d}x\)

kurtosis \(\kappa\) or \(\gamma_2\)

\( \displaystyle \operatorname{E}\Biggl[\biggl(\frac{X\!-\!\mu}{\sigma}\biggr)^{\!5}\Biggr] =\!\!\!\! \int\limits_{-\infty}^{\infty} {\color{maroon} \biggl(\frac{x\!-\!\mu}{\sigma}\biggr)^{\!5}} f(x) \,\mathrm{d}x\)

hyperskewness

A joint probability density function \(f\) of two random variables \(X\) and \(Y\) on a domain \(D\) is defined by the property that \(\iint_{D} f \,\mathrm{d}A = 1\,.\) For a region \(R,\) \(\operatorname{P}\bigl((X,Y) \in R\bigr) = \iint_R f \,\mathrm{d}A\) is the probability \((X,Y)\) is in \(R.\) The expected value of \(X\) or \(Y,\) also called the \(X\)-mean and \(Y\)-mean, are the first moments with respect to \(X\) and \(Y.\) \[ \operatorname{E}[X] = \iint_{\mathbf{R}^2} {\color{maroon} x} f \,\mathrm{d}A = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} {\color{maroon} x} f(x,y) \,\mathrm{d}x\,\mathrm{d}y \qquad \operatorname{E}[Y] = \iint_{\mathbf{R}^2} {\color{maroon} y} f \,\mathrm{d}A = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} {\color{maroon} y} f(x,y) \,\mathrm{d}x\,\mathrm{d}y \]