Question

The probability distribution of a random variable X is given below. Then, the standard deviation of X is.

• A. 5
• B. 11
• C. $\sqrt{11}$
• D. $\sqrt{5}$

Hint:

The standard deviation of a discrete random variable is the square root of the variance. The variance is calculated using the formula $\sigma^2 = E(X^2) - [E(X)]^2$. Always find the mean $E(X)$ first, then $E(X^2)$.

Answer 1

The Correct Option is C

Step 1: Find the value of k.
The sum of all probabilities in a probability distribution must be 1.
$\sum P(X=x_i) = 3k + k + k + 2k + k = 1$.
$8k = 1 \implies k = \frac{1}{8}$.
Step 2: Calculate the mean (expected value) of X, denoted by $E(X)$ or $\mu$.
$E(X) = \sum x_i P(X=x_i) = 2(3k) + 3(k) + 5(k) + 7(2k) + 12(k)$.
$E(X) = 6k + 3k + 5k + 14k + 12k = 40k$.
Substitute $k=1/8$: $E(X) = 40(\frac{1}{8}) = 5$.
Step 3: Calculate the expected value of $X^2$, denoted by $E(X^2)$.
$E(X^2) = \sum x_i^2 P(X=x_i) = 2^2(3k) + 3^2(k) + 5^2(k) + 7^2(2k) + 12^2(k)$.
$E(X^2) = 4(3k) + 9(k) + 25(k) + 49(2k) + 144(k) = 12k+9k+25k+98k+144k = 288k$.
Substitute $k=1/8$: $E(X^2) = 288(\frac{1}{8}) = 36$.
Step 4: Calculate the variance of X, denoted by $\text{Var}(X)$ or $\sigma^2$.
$\text{Var}(X) = E(X^2) - [E(X)]^2$.
$\text{Var}(X) = 36 - 5^2 = 36 - 25 = 11$.
Step 5: Calculate the standard deviation of X, denoted by $\sigma$.
Standard Deviation, $\sigma = \sqrt{\text{Var}(X)} = \sqrt{11}$.