Question

Which of the following are not the probability distribution of a random variable ?
A.

X	0	1	2
P(X)	0.4	0.4	0.2

B.

X	0	1	2	3	4
P(X)	0.4	0.4	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	0.05

E.

X	0	1	2
P(X)	\(\frac{25}{36}\)	\(\frac{10}{36}\)	\(\frac{1}{36}\)

Choose the correct answer from the options given below :

• A. A and E only
• B. B, C and D only
• C. A, D and E only
• D. C, A and D only

Answer 1

The Correct Option is B

To determine which distributions are not probability distributions of a random variable, we must ensure that two conditions are met for each distribution: 1. The probabilities must be non-negative. 2. The sum of the probabilities must equal 1.

Let's evaluate each case:

1. A:

   X	0	1	2
   P(X)	0.4	0.4	0.2

   All probabilities are non-negative, and their sum is \(0.4 + 0.4 + 0.2 = 1\). This is a valid probability distribution.
2. B:

   X	0	1	2	3	4
   P(X)	0.4	0.4	0.2	-0.1	0.3

   Contains a negative probability (-0.1) which is not allowed. Thus, this is not a valid probability distribution.
3. C:

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

   The sum of probabilities is \(0.6 + 0.1 + 0.2 = 0.9\) which is not equal to 1. Hence, this is not a valid probability distribution.
4. D:

   Z	3	2	1	0	-1
   P(Z)	0.3	0.2	0.4	0.1	0.05

   Although all probabilities are non-negative, their sum is \(0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05\) which is greater than 1. Therefore, this is not a valid probability distribution.
5. E:

   X	0	1	2
   P(X)	\(\frac{25}{36}\)	\(\frac{10}{36}\)	\(\frac{1}{36}\)

   All probabilities are non-negative and their sum is \(\frac{25}{36} + \frac{10}{36} + \frac{1}{36} = \frac{36}{36} = 1\). This is a valid probability distribution.

Based on the analysis, the distributions that are not valid probability distributions are from B, C, and D. Thus, the correct answer is: B, C and D only.