Distribution functions

Summary

The cumulative distribution function (cdf)

• If \(X\) is a random variable, then for a given real number \(x\) we define the cumulative distribution function of \(X\) as the probability that \(X\) will take a value less than or equal to \(x\) :

\[F\left( x \right)=P(X≤x)\]

The probability mass function (pmf)

• If \(X\) is a discrete random variable, then for a given real number \(x\) we define the probability mass function of \(X\) as the probability that \(X\) will take precisely the value \(x\) :

\[f\left( x \right)=P(X=x)\]

• If \(x\) is not in the range of \(X\) then \(f\left( x \right)=0\) .

The probability density function (pdf)

• If \(X\) is a continuous random variable, then \(P\left( X=x \right)=0\) for all real numbers \(x\) .
• We define the probability density function of \(X\) as
• \(f\left( x \right)= \frac{dF(x)}{dx}\)
• (if the derivative exists)
• If \(X\) is a continuous random variable with probability density function \(f(x)\) then

\[P\left( a≤X≤b \right)=\int_{a}^{b}{ f\left( x \right)dx }\]