Cumulative Distribution Function (CDF)

CDF (Cumulative Distribution Function)

The cumulative distribution function (cdf) $F (x)$ for a continuous rv with PDF $f (x)$ is defined for every number $x$ by:

F (x) = P (X \leq x) = \int_{- \infty}^{x} f (t) d t

For each $x$ , $F (x)$ is the area under the density curve to the left of $x$ .

Using

F (x)

to compute probabilities

Let $X$ be a continuous rv with pdf $f (x)$ and corresponding cdf $F (x)$ . Then for any number $a$ :

P (X > a) = 1 - F (a)

and for any two numbers $a, b$ with $a < b$ :

P (a \leq X \leq b) = F (b) - F (a)

These come from the fact that $P (X^{'}) = 1 - P (X)$ for (1) and for (2) just using this idea in the expansion of the areas:

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