2.2c.1 Probability Density Function

Probability Density Function (PDF)

The probability distribution of a continuous random variable $X$ is fully described by a non-negative probability density function (PDF) $f_X:\mathcal{X}\to [0,\infty )$, which illustrates the density around each value $x\in \mathcal{X}$. In particular, the probability that the value of $X$ falls within an interval $[a,b]$ over the real line is given as
$\mathbb{P}(a\le X\le b)={\int }_a^b{f}_X(x)dx,$
and can be interpreted as the area under the graph of the PDF.

The cumulative probability that $X$ takes on a value less than or equal to $x$ in the PDF is given by the non-decreasing cumulative distribution function (CDF) $F_X:\mathbb{R}\to [0,1]$ , formally defined as
$F_X(x)=\mathbb{P}(X\le x)={\int }_{-\infty }^x{f}_X(t)dt,$
implying that the PDF is the rate of change of the CDF, $f_X(x)=\frac{d}{dx}{F}_X(x)$.

PDFs Measure Density $x$, $\mathbb{P}(X=x)={\int }_x^x{f}_X(x)dx=0$. However, an alternative to this is to select the interval $[x,x+\delta ]$ for a very small length $\delta$, and compute ${\int }_x^{x+\delta }{f}_X(x)dx\approx f_X(x)\cdot \delta .$

For an individual point

Properties

(Event Conditioning) If $A\in \mathcal{F}\text{ s.t. }\mathbb{P}(X\in A)>0$, then $f_{X\mid A}(x)=\{\begin{array}{ll}\frac{f_X(x)}{\mathbb{P}(X\in A)}& x\in A,\\ 0& \text{o.w.}\end{array}$

(R.V. Conditioning) $f_{X\mid Y}(x\mid y)=$

(Joint PDFs) $f_{X,Y}(x,y)=f_Y(y)f_{X\mid Y}(x\mid y)$.

(Marginality) $f_X(x)={\int }_{-\infty }^{\infty }{f}_Y(y)f_{X\mid Y}(x\mid y)dy$.

(Continuous Bayes’) $f_{X\mid Y}(x\mid y)=\frac{f_X(x)f_{Y\mid X}(y\mid x)}{f_Y(y)}=\frac{f_X(x)f_{Y\mid X}(y\mid x)}{\int f_X(t)f_{Y\mid X}(y\mid t)dt}.$

(Independence) R.V.’s $X,Y$ are independent iff $f_{X,Y}(x,y)=f_X(x)\cdot y_Y(y)$ $\forall x,y$.

(Joint CDFs)

(Sum Convolution) If $W=X+Y$, $f_W(w)={\int }_{-\infty }^{\infty }{f}_X(x)f_Y(w-x)dx$.