Statistics/Distributions/Continuous

A continuous statistic is a random variable that does not have any points at which there is any distinct probability that the variable will be the corresponding number.

General Properties

Cumulative Distribution Function

A continuous random variable, like a discrete random variable, has a cumulative distribution function. Like the one for a discrete random variable, it also increases towards 1. Depending on the random variable, it may reach one at a finite number, or it may not. The cdf is represented by a capital F.

Probability Distribution Function

Unlike a discrete random variable, a continuous random variable has a probability density function instead of a probability mass function. The difference is that the former must integrate to 1, while the latter must have a total value of 1. The two are very similar, otherwise. The pdf is represented by a lowercase f.

Special Values

Let R be the set of points of the distribution.

The expected value for a continuous variable X with probability density function f is defined as $\underset{R}{\int }xf(x)dx$.

More generally, the expected value of any continuously transformed variable g(X) with probability density function f is defined as $\underset{R}{\int }g(x)f(x)dx$.

The mean of a continuous or discrete distribution is defined as $E[X]$.

The variance of a continuous or discrete distribution is defined as $E[(X-E[X{]}^2)]$.

Expectations can also be derived by producing the Moment Generating Function for the distribution in question. This is done by finding the expected value $E[\exp (tX)]$. Once the Moment Generating Function has been created, each derivative of the function gives a different piece of information about the distribution function.

$\frac{dE[\exp (tX)]}{dt}$ = mean
$\frac{d^{2}E[\exp (tX)]}{d{t}^2}$ = variance
$\frac{d^{3}E[\exp (tX)]}{d{t}^3}$ = skewness
$\frac{d^{4}E[\exp (tX)]}{d{t}^4}$ = kurtosis