For the following probability distribution:
| X | 3 | 4 | 5 |
| P(X) | 0.5 | 0.2 | 0.3 |
The mean, variance and standard deviation respectively are:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3.8, 0.76 and 0.87
The mean is \(E(X) = \sum x\,P(x)\):
\(E(X) = 3(0.5) + 4(0.2) + 5(0.3)\)
\(= 1.5 + 0.8 + 1.5 = 3.8\)
For the variance, \(\mathrm{Var}(X) = E\left(X^2\right) - [E(X)]^2\):
\(E\left(X^2\right) = 9(0.5) + 16(0.2) + 25(0.3)\)
\(= 4.5 + 3.2 + 7.5 = 15.2\)
\(\mathrm{Var}(X) = 15.2 - 14.44 = 0.76\)
The standard deviation is the square root of the variance:
\(\sigma = \sqrt{0.76} \approx 0.87\)
So the mean, variance and standard deviation are 3.8, 0.76 and 0.87.