Cumulative distribution functions and expected values

Cumulative distribution function of a continuous random variable Whereas for a discrete random variable we sum probabilities up to a point x to obtain the cumulative distribution function , in the continuous case we integrate.

Definition and properties of the cumulative distribution function The CDF of a continuous random variable with density function f(x) is The function F(x) takes values in [0,1], i.e. , and is non-decreasing.

Examples For X with pdf, the CDF is given by

Angles corresponding to an imperfection on a tire For this example, In general, for a uniform rv on [A,B],

Using F(x) to compute probabilities Let X be a continuous random variable with pdf f(x) and cdf F(X). Then for any number a, and for any two numbers a and b,

Obtaining f(x) from F(x) If X is a continuous rv with pdf f(x) and cdf F(x), then at every x at which the derivative exists, . For ,

Percentiles of a continuous distributon Let 0<p<1. The (100p)th percentile of a rv, denoted by , is defined by The median is the 50th percentile. The text denotes the median by .

Median of a symmetric distribution A continuous distribution whose pdf is symmetric has median equal to the point of symmetry.

Expected values of continuous random variables Expected values of continuous random variables are computed in a manner similar to the discrete case, with integrals replacing sums.

Definition The expected value or mean of a continuous rv X with pdf f(x) is computed as For a function h(X),

Definition of variance The variance of a continuous random variable with pdf f(x) and mean value is The variance may also be computed using