## 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 x^{t}) 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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