2-Probability mass function 

A random variable that can take on at most a countable number of possible values

is said to be discrete. For a discrete random variable X, 

we define the probability mass function p(a) of X by 𝑝(π‘Ž)=𝑃{𝑋 =π‘Ž}

The probability mass function p(a) is positive for at most a countable number

of values of a. That is, if X must assume one of the values  . . . ,

then

𝑝()≥0  for 𝑖 =1,2,β‹―

𝑝(π‘₯)=0 for all ather values of π‘₯

Since X must take on one of the values  we have 

It is often instructive to present the probability mass function

in a graphical format by plotting p() on the y-axis against on the x-axis.

For instance, if the probability mass function of X is

We can represent this function graphically as: 

Similarly,  a  graph  of  the  probability  mass  function  of  the

random variable representing the sum when two dice are rolled looks like

The  probability  mass  function  of  a  random  variable  X  is  given  by

 = 0, 1, 2, β‹―, where λ is some positive value.

Find

(a) P{X = 0}

and

(b) P{X > 2}.

Since =1, we have 

which, because 

Hence, 

(a)𝑃{𝑋 =0}= 

(b)𝑃{𝑋 >2}=1−𝑃{𝑋 ≤2}=1−𝑃{0}−𝑃{𝑋 =1}−𝑃{𝑋 =2}

The cumulative distribution function F can be expressed in terms of p(a) by

If X is a discrete random variable whose possible values are 

 then the distribution function F of X is a step function.

That is, the value of F is constant in the intervals

and then takes a step (or jump) of size 

instance, if X has a probability mass function given by

then its cumulative distribution function is

This function is depicted graphically as

Note that the size of the step at any of the values 1, 2, 3, and 4

is equal to the probability that X assumes that particular value.