CDF vs PDF-Difference between CDF and PDF

This page CDF vs PDF describes difference between CDF(Cumulative Distribution Function) and PDF(Probability Density Function).

A random variable is a variable whose value at a time is a probabilistic measurement. It is mapping from the sample space to the set of real number.

CDF-Cumulative Distribution Function

CDF i.e. Cumulative Distribution Function of a random variable X is defined as
Fx(x) = P (X <= x)

Properties of CDF are as follows:
•  0 <=Fx(x)<= 1
•  Fx(x) is non decreasing function
•  lim Fx(x) = 0 (where x -> -∞) and lim Fx(x) =1 (where x -> +∞)
•  Fx(x) is always continuous from right that is F(x+ε) = F(x)
•  P(a<X<=b) = Fx(b)-Fx(a)
•  P(X=a) = Fx(a)-Fx(a')

Following are the important features of CDF:
• For discrete random variable Fx(x) is a stair case function.
• For continuous random variable CDF is continuous.

Refer CCDF basics.

PDF-Probability Density Function

PDF i.e. Probability Density Function of a random variable X is defined as the derivative of CDF that is
Fx(x) = d/dx(Fx(x))

Properties of PDF are as follows:
•  Fx(x) >= 0
•  Integrate(from -∞ to +∞)Fx(x) dx = 1, total probability
•  Integrate(from a+ to b-)Fx(x) dx = P (a<X<=b)
•  Fx(x) = Integrate(from -∞ to xt) Fx(u)du

For discrete random variables it is more common to define the probability mass function (PMF) which is defined as PMF = {Pi}
Where, Pi = P (X = xi)

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