The random variable X has Cumulative Distribution Function F

The random variable X has Cumulative Distribution Function F_X (x) = {0, x

Solution

baf(x)dx=Pr[aXb]

For a discrete distribution, the pdf is the probability that the variate takes the value x.

f(x)=Pr[X=x]

The cumulative distribution function (cdf) is the probability that the variable takes a value less than or equal to x. That is

F(x)=Pr[Xx]=

For a continuous distribution, this can be expressed mathematically as

F(x)=xf()d

For a discrete distribution, the cdf can be expressed as

F(x)=xi=0f(i)

Probability distributions are typically defined in terms of the probability density function. However, there are a number of probability functions used in applications.

Probability Density Function

For a continuous function, the probability density function (pdf) is the probability that the variate has the value x. Since for continuous distributions the probability at a single point is zero, this is often expressed in terms of an integral between two points. baf(x)dx=Pr[aXb] For a discrete distribution, the pdf is the probability that the variate takes the value x. f(x)=Pr[X=x] The cumulative distribution function (cdf) is the probability that the variable takes a value less than or equal to x. That is F(x)=Pr[Xx]= For a continuous distribution, this can be expressed mathematically as F(x)=xf()d For a discrete distribution, the cdf can be expressed as F(x)=xi=0f(i)