What Is Cumulative Distribution Function
Cumulative distribution function Definition - what it is A cumulative distribution function (CDF) for a real-valued random variable ---- is the function ---- ---- ( ---- ) that gives the probability that ---- takes a value less than or equal to ---- : ---- ---- ( ---- ) = ---- ( ---- - ---- ) . Key properties of a CDF Nondecreasing: ---- ---- ( ---- ) never decreases as ---- increases. Right-continuous: ---- ---- ( ---- ) = lim - ---- - ---- ---- ---- ( ---- ) . Limits: lim - ---- - - - ---- ---- ( ---- ) = 0 and lim - ---- - + - ---- ---- ( ---- ) = 1 . Relation to PDF and PMF Continuous case: If ---- has a probability density function ---- ( ---- ) , then the CDF is the integral of the PDF: ---- ---- ( ---- ) = - - - ---- ---- ( ---- ) - ---- ---- , and ---- ( ---- ) = ---- ---- ---- ---- ---- ( ---- ) where the derivative exists. Discrete case: For a discrete random variable, the CDF is a step function; each jump at a point ---- ---- equals the probability ---- ( ---- = ---- ---- ) . How to use a CDF Compute probabilities over intervals: ---- ( ---- < ---- - ---- ) = ---- ---- ( ---- ) - ---- ---- ( ---- ) . Describe the full distribution: The CDF uniquely characterizes a distribution (continuous, discrete, or mixed) and is useful for plotting, quantiles, and simulations. Quick example (normal): For a standard normal random variable ---- , the CDF is ---- ---- ( ---- ) = - - - ---- 1 2 ---- ---- - ---- 2 / 2 - ---- ---- , which gives the probability ---- ( ---- - ---- )