The probability distribution of the random variable X is given by
| X | 0 | 1 | 2 | 3 |
| P(X) | 0.2 | k | 2k | 2k |
The variance of the random variable X is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{764}{625}\)
The probabilities add up to 1:
\(0.2 + k + 2k + 2k = 1\)
\(5k = 0.8,\quad k = 0.16\)
Mean:
\(E(X) = 0(0.2) + 1(k) + 2(2k) + 3(2k) = 11k\)
\(E(X) = 11 \times 0.16 = 1.76\)
Mean of the squares:
\(E(X^2) = 0(0.2) + 1(k) + 4(2k) + 9(2k) = 27k\)
\(E(X^2) = 27 \times 0.16 = 4.32\)
By the variance formula \(\mathrm{Var}(X) = E(X^2) - [E(X)]^2\):
\(\mathrm{Var}(X) = 4.32 - (1.76)^2\)
\(= 4.32 - 3.0976 = 1.2224\)
As a fraction:
\( \displaystyle 1.2224 = \frac{12224}{10000} = \frac{764}{625} \)