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) | \(\displaystyle\frac{25}{36}\) | \(\displaystyle\frac{10}{36}\) | \(\displaystyle\frac{1}{36}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B, C and D only
A table is a probability distribution only if every probability is non-negative and the probabilities add up to 1. Adding each row:
\(\text{A: } 0.4 + 0.4 + 0.2 = 1\)
\(\text{B: } 0.4 + 0.4 + 0.2 - 0.1 + 0.3 = 1.2\)
\(\text{C: } 0.6 + 0.1 + 0.2 = 0.9\)
\(\text{D: } 0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05\)
\( \displaystyle \text{E: } \frac{25}{36} + \frac{10}{36} + \frac{1}{36} = \frac{36}{36} = 1 \)
So the sums are A: 1, B: 1.2 (with a negative value −0.1), C: 0.9, D: 1.05 and E: 1, and only A and E have all values non-negative with sum 1.
Checking the statements
(A) P(X) = 0.4, 0.4, 0.2 — Incorrect. It is a valid probability distribution (all values non-negative, sum 1), so it is not one of the tables asked for.
(B) P(X) = 0.4, 0.4, 0.2, −0.1, 0.3 — Correct. Not a distribution: P(X = 3) = −0.1 is negative (and the sum is 1.2).
(C) P(Y) = 0.6, 0.1, 0.2 — Correct. Not a distribution: the sum is 0.9, not 1.
(D) P(Z) = 0.3, 0.2, 0.4, 0.1, 0.05 — Correct. Not a distribution: the sum is 1.05, not 1.
(E) P(X) = 25/36, 10/36, 1/36 — Incorrect. It is a valid probability distribution (all values non-negative, sum 1), so it is not one of the tables asked for.