Question

Explain why the following is or is not a valid probability distribution for the discrete random variable  x.

x	1	0	1	2	3
p(x)	.1	.2	.3	.3	.1

Answer 1

Solution

Here we want to prove that the given probability mass function (pmf) is not valid. Given that,

X	1	0	1	2	3
p(x)	0.1	0.2	0.3	0.3	0.1

Now consider the below definition,

If X is a discrete random variable with distinct values x₁,x₂,......,x_n ,..., then the function p(x) defined as:

is called the probability mass function of the random variable X.

Now when we look into the given table probabilitu mass function for x=1 is not unique. That is for x=1 there are two different pmf is given (0.1 and 0.3). Which means there are different probability for the occurance of same event x=1. It is not possible . Hence we can say that it is not a valid probability distribution for the discrete random variable X.