Question

State which of the following are not the probability distribution of a random variable. Give reasons for your answer.
(a)

X	0	1	2
P(X)	0.4	0.4	0.2

(b)

X	0	1	2	3	4
P(X)	0.1	0.5	0.2	-0.1	0.3

(c)

Y	-1	0	1
P(Y)	0.6	0.1	0.2

(d)

Z	3	2	1	0	-1
P(Z)	0.3	0.2	0.4	0.1	0.0

Answer 1

Hint: First, before proceeding for this, we must know that the probability distribution function always gives the total of 1. Then, by using the above mentioned condition, we will check option by option to get the result whether the given distribution function is valid or not. Then, by using the condition stated, we get which option is not a correct distribution table.

Complete step by step answer:
In this question, we are supposed to find which of the following are not the probability distribution of a random variable and also give reasons for the answer.
So, before proceeding for this, we must know that the probability distribution function always gives the total of 1.
Then, by using the above mentioned condition, we will check option by option to get the result whether the given distribution function is valid or not.
So, by starting with the first option, we get the sum of the probabilities as given in the table as:

X	0	1	2
P(X)	0.4	0.4	0.2

0.4+0.4+0.2=1
So, we get the total probability as 1 which proves it is a valid distribution function.
Then, by moving with the second option, we get the sum of the probabilities as given in the table as:

X	0	1	2	3	4
P(X)	0.1	0.5	0.2	-0.1	0.3

0.1+0.5+0.2-0.1+0.3=1
So, we get the total probability as 1 which proves it is a valid distribution function.
Then, by moving with the third option, we get the sum of the probabilities as given in the table as:

Y	-1	0	1
P(Y)	0.6	0.1	0.2

0.6+0.1+0.2=0.9
So, we get the total probability as 0.9 which proves it is not a valid distribution function.
Then, by moving with the fourth option, we get the sum of the probabilities as given in the table as:

Z	3	2	1	0	-1
P(Z)	0.3	0.2	0.4	0.1	0.0

0.3+0.2+0.4+0.1+0.0=1
So, we get the total probability as 1 which proves it is a valid distribution function.
Hence, option (c) is correct.

Note:
Now, to solve these types of questions we need to be careful with the given tables. The first two tables are defined with the random variable X and the third and fourth table is defined with random variables Y and Z respectively. However, there is no relation of the validity of the probability distribution table with the random variable and they are just given differently for making confusion in the solver's mind.