Home page for accesible maths 2.4 Continuous random variables 2.4 Continuous random variables An illuminating idea

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2.4.1 The probability density function

Recall (Equation (2.1)) that the CDF, $F_{R}(r)$ , of a discrete random variable, $R$ , is the sum of all of the relevant probabilities, $\sum_{i=-\infty}^{\operatorname{int}(r)}p_{R}(i)$ .

Analogously the (probability) density function, pdf, $f_{X}(x)$ of a continuous random variable, $X$ , is defined as

\displaystyle f_{X}(x)=\frac{d}{dx}F_{X}(x),

so that it satisfies

\displaystyle F_{X}(x)=\int_{-\infty}^{x}f_{X}(s)\,\mathrm{d}s.

(2.2)

For example

\displaystyle\operatorname{\mathsf{P}}\left({X\leq 10}\right)=F_{X}(10)=\int_{% -\infty}^{10}f_{X}(s)\,\mathrm{d}s.

Note that $s$ is a dummy variable; one could use any letter for the integrand in (2.2), except $x$ .

Two frequent student mistakes in Math230 are

1.

Writing $F_{X}(x)=\int f_{X}(x)\,\mathrm{d}x$ ; i.e. writing the indefinite integral instead of a definite integral.
2.

Writing $F_{X}(x)=\int_{-\infty}^{x}f_{X}(x)\,\mathrm{d}x$ , which is mathematical nonsense.

Figure 2.3: Link, Caption: The probability density function corresponding to the cdf shown in Figure 2.2 (Link).

Figure 2.3 (Link) shows the pdf for the continuous random variable for which the cdf is shown in Figure 2.2 (Link). Notice that the pdf is zero in regions where there are no outcomes, in this example for $x\leq 0$ . Note also that it exceeds $1$ in some places, so it cannot be interpreted as a probability despite some of the mathematical properties of $f_{X}(x)$ that we shall see below being very similar to those of the probability mass function.

Properties of $f_{X}(x)$ :

1.

Positivity: $f_{X}(x)\geq 0$ for all $x$ , Quiz: Why?
2.

Unit-integrability: $\int_{-\infty}^{\infty}f_{X}(x)\,\mathrm{d}x=1$ . Quiz: Why?

The probability that an observation on a continuous random variable $X$ lies in the interval $(a,b]$ may be calculated as the area under the curve of $f_{X}(x)$ between $x=a$ and $x=b$ . In the folllowing figure, $a=-0.5$ and $b=2$ .

Unnumbered Figure: Link

More formally:

\displaystyle\operatorname{\mathsf{P}}\left({a<X\leq b}\right)=F_{X}(b)-F_{X}(% a)=\int_{-\infty}^{b}f_{X}(s)\,\mathrm{d}s-\int_{-\infty}^{a}f_{X}(s)\,\mathrm% {d}s=\int_{a}^{b}f_{X}(s)\,\mathrm{d}s.

The above can be extended to the probability of any set $A$ given by

\displaystyle\operatorname{\mathsf{P}}\left({X\in A}\right)=\int_{s\in A}f_{X}% (s)\,\mathrm{d}s.

(2.3)

e.g. $\operatorname{\mathsf{P}}\left({|X|>1}\right)=\int_{-\infty}^{-1}f_{X}(s)\,% \mathrm{d}s+\int_{1}^{\infty}f_{X}(s)\,\mathrm{d}s$ .