Question

difference of cumulative and probability distribution

Answer 1

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.



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)