The probability distribution of a discrete random variable is given below:
| \( X \) | 1 | 2 | 3 | 4 | 5 |
| P(X) | \( \displaystyle\frac{1}{5} \) | m | \( \displaystyle\frac{2}{5} \) | 2m | 3m |
Which of the following are correct?
A. \( P(2)=\displaystyle\frac{1}{15} \)
B. \( P(4)=\displaystyle\frac{2}{15} \)
C. \( P(X\lt 2)=\displaystyle\frac{1}{5} \)
D. \( P(X\ge 4)=\displaystyle\frac{1}{5} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A, B and C only
The probabilities must add up to 1:
\( \displaystyle \frac{1}{5}+m+\frac{2}{5}+2m+3m=1 \)
\( \displaystyle 6m=\frac{2}{5} \)
\( \displaystyle m=\frac{1}{15} \)
So m = 1/15, and the distribution is P(1) = 1/5, P(2) = 1/15, P(3) = 2/5, P(4) = 2/15 and P(5) = 3/15.
Checking the statements
(A) \( P(2)=\displaystyle\frac{1}{15} \) — Correct. P(2) = m = 1/15.
(B) \( P(4)=\displaystyle\frac{2}{15} \) — Correct. P(4) = 2m = 2/15.
(C) \( P(X\lt 2)=\displaystyle\frac{1}{5} \) — Correct. The only value below 2 is X = 1, and P(1) = 1/5.
(D) \( P(X\ge 4)=\displaystyle\frac{1}{5} \) — Incorrect. P(4) + P(5) = 2/15 + 3/15 = 1/3.