Question

What is difference between probability density function and cumulative distribution function ??

Answer 1

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CDF-Cumulative Distribution Function
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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.

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PDF-Probability Density Function
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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)