Distributions and Densities

A cumulative distribution function F $\left(\vphantom{ x}\right.$ x $\left.\vphantom{ x}\right)$ of a random variable X is the function F $\left(\vphantom{ x}\right.$ x $\left.\vphantom{ x}\right)$ = P $\left(\vphantom{ X\leq x}\right.$ X≤x $\left.\vphantom{ X\leq x}\right)$ , the probability that X≤x. If F(x) has a derivative f (x), then f (x) is nonnegative and is called the probability density function of x. The inverse distribution function G(α) satisfies G $\left(\vphantom{ F\left( x\right) }\right.$ F $\left(\vphantom{ x}\right.$ x $\left.\vphantom{ x}\right)$ $\left.\vphantom{ F\left( x\right) }\right)$ = x and F $\left(\vphantom{ G\left( \alpha \right) }\right.$ G $\left(\vphantom{ \alpha }\right.$ α $\left.\vphantom{ \alpha }\right)$ $\left.\vphantom{ G\left( \alpha \right) }\right)$ = α. The Scientific Notebook names for these functions are obtained by adding Dist, Den, or Inv to the name of the distribution. For example, $\func$ NormalDist, $\func$ NormalDen, and $\func$ NormalInv are the three functions for the normal distribution. Such function names will automatically turn gray when typed in mathematics mode.

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