Probability mass function and density function

Random variables are always generated with a particular pattern of probability attached to them. Thus, based on the pattern of probability for the different values of random variable, we can distinguish them. Once we know these probability distributions and their properties, and if any random variable fits in a probability distribution, so we will see what is probability mass function and probability density function

A probability mass function (PMF)also called a frequency function gives you probabilities for discrete random variables. “Random variables” are variables from experiments like dice rolls, choosing a number out of a hat, or getting a high score on a test. The “discrete” part means that there’s a set number of outcomes. For example, you can only roll a 1, 2, 3, 4, 5, or 6 on a die.

for example pmf for binomial distribution...

Probability Mass Function

Probability mass function is a function of defining a discrete probability distribution,whose domain is discrete.sometimes it also known as discrete density function.
  Let X be a discrete random variable.let X1,X2,X3,...Xn be the possible values of X.with each possible outcome xi we associate a number P(xi)=P(X=xi) is called probability of xi.the number P(xi) i=1,2,3,...n which satisfies following conditions
P(xi) ≥ 0
ΣP(xi)=1
Then the function P(xi) is called probability mass function.

Consider 
   X = 1.      2.     3
P(x)= 0.2  0.3   0.5
Here all P(x) are non-negative i.e P(x) ≥ 0
 Σ P(x)=0.2+0.3+0.5=1
Hence given function is probability mass function.

Probability density Function

Probability density function is a function of defining a continuous probability distribution,whose domain is continuous. it also known as  density function.
  Let X be a continuous random variable. f(x) is continuous function of X, X has range [a,b] which satisfies following conditions
f(xi) ≥ 0. For all x
∫f(x)dx=1...(integration a to b)
Then the function f(x) is called probability density function.

 Consider the following example,



 Hence given f(x) is probability density function.
 
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