Mathematics · Ch 11 — Probability Distributions

Probability density function

11.4.2

Probability density function

Definition 11.6 (Probability density function). A non-negative real-valued function $f(x)$ is a probability density function (pdf) of a continuous random variable $X$ if, for every $a\le b$ in the range of $X$ ,

P(a\le X\le b)=\int_a^b f(x)\,dx

— i.e. probability over an interval is the area under the curve $y=f(x)$ between $x=a$ and $x=b$ .

Theorem 11.2 (characterisation, without proof). $f(\cdot)$ is a pdf for some continuous random variable $X$ if and only if (i) $f(x)\ge0$ for every $x$ , and (ii) $\displaystyle\int_{-\infty}^{\infty}f(x)\,dx=1$ (total area under the curve is exactly $1$ , the discrete " $\sum f(x_k)=1$ " condition translated into area form).

Note

Taking $a=b$ in the defining formula gives $P(X=a)=\int_a^a f(x)\,dx=0$ — recovering Definition 11.5 directly: the probability that a continuous $X$ equals any one particular value is always zero, however large $f(a)$ itself may be. $f(a)$ is a density, not a probability. …

Figure 11.11A probability density function f(x) with the area under the curve between x = a and x = b shaded, representing P(a <= X <= b) as the integral of f(x) from a to b

Fig. 11.11 — A probability density function f(x) with the area under the curve between x = a and x = b shaded, representing P(a <= X <= b) as the integral of f(x) from a to b

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. A probability density function f(x) with the area under the curve between x = a and x = b shaded, representing P(a <= X <= b) as the integral of f( …

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