Recall that a probability density function (PDF) \(f\) of a single random variable \(X\) is defined by the property that \(\int_{\mathbf{R}} f(x) \,\mathrm{d}x = 1\,.\) 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 \(\operatorname{P}(X \in I) = F(b) - F(a).\) Common examples of probability density functions include:
The expected value of \(X\), denoted \(\operatorname{E}[X],\) also 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 (raw) moment of the PDF. Higher moments of a PDF are generally measured from the mean and are referred to as central moments. For example, TK standardized or normalized
A joint probability density function \(f\) of two random variables \(X\) and \(Y\) is defined by the property that \(\iint_{\mathbf{R^2}} f \,\mathrm{d}A = 1\,.\) 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 \(Y.\) \[ \operatorname{E}[X] = \iint_{\mathbf{R}^2} X f \,\mathrm{d}A = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} X f(X,Y) \,\mathrm{d}X\,\mathrm{d}Y \qquad \operatorname{E}[Y] = \iint_{\mathbf{R}^2} Y f(X,Y)\,\mathrm{d}A \] These are the first moments ;) Higher moments also have interpretations in statistics and probability.