The probability distribution table of a random variable X is given below.
| Xi | P(Xi) |
|---|---|
| 0 | 1/8 |
| 1 | a |
| 2 | b |
| 3 | 1/8 |
If \( \mathrm{E(X)}=\displaystyle\frac{3}{2} \), then the values of a and b are:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{3}{8} \) and \( \displaystyle\frac{3}{8} \) respectively
The probabilities add up to 1:
\( \displaystyle \frac{1}{8}+a+b+\frac{1}{8}=1 \)
\( \displaystyle a+b=\frac{3}{4}=\frac{6}{8} \)
The mean is \( E(X)=\sum x_ip_i \):
\( \displaystyle 0\cdot\frac{1}{8}+1\cdot a+2\cdot b+3\cdot\frac{1}{8}=\frac{3}{2} \)
\( \displaystyle a+2b=\frac{3}{2}-\frac{3}{8}=\frac{9}{8} \)
Subtracting the first equation from the second:
\( \displaystyle b=\frac{9}{8}-\frac{6}{8}=\frac{3}{8} \)
\( \displaystyle a=\frac{3}{4}-\frac{3}{8}=\frac{3}{8} \)
Shortcut: All four options satisfy a + b = 3/4, so the total-probability condition alone cannot decide. Only the mean condition a + 2b = 9/8 separates them: Option (1) gives 3/8 + 6/8 = 9/8, while Options (2), (3) and (4) give 10/8, 8/8 and 11/8.