cumulative distribution function

 

In statistics and probability, the cumulative distribution function, abbreviated cdf, of a real-valued variable X is a function that, when evaluated at X=x, gives the probability that X has a value less than or equal to x. By convention, cumulative distribution functions are denoted by upper case letters, often F. The cdf F for random variable would be F_X. So, by definition:

F_X(x) = P(X \le x)

For a discrete probability distribution over X, the cumulative distribution function at X=x is the sum of the probability mass function values of X starting with the first X having non-zero probability up to and including x.

F_X(x) =\displaystyle\sum_{x_i \le x}^{x}f_X(x_i) = \displaystyle\sum_{x_i \le x}^{x}P(X = x_i)

If the random variable X is represented by a continuous distribution, the cumulative distribution function at X=x gives the area under the probability density function from minus infinity to x. This is equivalent to the integral of the probability distribution function from minus infinity to x:

\displaystyle\int_{-\infty}^{x}f_X(t)dt

The cumulative distribution function is non-decreasing and right-continuous. Because probabilities must be between 0 and 1, and the totally probability of sample space is 1:

\displaystyle\lim_{x\to-\infty}F(x)=0, \quad\displaystyle\lim_{x\to\infty}F(x)=1

 

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