Topics:

Cumulative distribution functions

Continuous random variables and their probability densities (MAT2572RVsAndTheirPDsNotesContinuousCase)

Cumulative distribution function for a continuous RV = the antiderivative of the probability density (being careful of domain)

Brief notes on cumulative distribution functions:

Definition: For a random variable $X$, the cumulative distribution function (cdf) $F(x)$ is a function with domain the set of all real numbers, such that

$F(x) = P(X\le x)$

Note: Capital $X$ is the name of  the random variable. Lower-case $x$ is the input to the function, that is, it represents some real number.

Facts about cdfs:

• For any type of random variable (discrete or continuous), $F(x)$ is a nondecreasing function.

• $\displaystyle \lim_{x\rightarrow -\infty}F(x) = 0$ and $\displaystyle \lim_{x\rightarrow \infty}F(x) = 1$

• If $X$ is a discrete (finite or infinite) RV, $F(x)$ is a step function with jumps at the possible values of $X$.

• The use of the cdf: For any real numbers $a$ and $b$, $a\le b$,

$P(a \le X < b) = F(b) – F(a)$

Often (especially for continuous RVs) it is much easier to compute probabilities using the cdf rather than the pdf!