Cumulative Distribution Function

## Description

The Cumulative Distribution Function(CDF) is the probability that a distribution function of X will take a value less than or equal to x. The CDF of a continuous random variable X is expressed as

(1)
\begin{align} P( X \leq x)= \int_{-\infty}^{x} f(t)dt. \end{align}

The Sage cell below calculates the CDF for the exponential distribution when $\lambda=4$ and $X<.75$.

## Sage Cell

#### Code

l=4
f(X)=l * exp( -l* x)
CDF=integral(f(X),x,0,.75)
CDF


## Options

#### CDF for Discrete Distributions

The CDF of a discrete probability distribution function is

(2)
\begin{align} E(X<x) = \sum_{u=0}^{x} f(u) \end{align}

The Sage cell below calculates the CDF of the poisson distribution when $\lambda=15$ and $x<12$.

#### Code

lamb = 15
x=12
var('k')
f(k)=exp( -lamb) * (( lamb^k ) / ( factorial(k)))
CDF=sum(f(k) , k, 0, x)
numerical_approx(CDF)


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