Question

A random variable X has the distribution: $P(X=1,2,3,4) = 0.1, 0.2, 0.3, 0.4$. The mean and standard deviation are:

• A. 2 and 3
• B. 3 and 1
• C. 3 and $\sqrt{1}$
• D. 3 and 1

Hint:

Standard deviation is always the positive square root of the variance.

Answer 1

The Correct Option is B

Step 1: Formula
Mean $E(X) = \sum x_i P_i$. Variance $V(X) = \sum x_i^2 P_i - [E(X)]^2$.

Step 2: Analysis
Mean $= (1 \times 0.1) + (2 \times 0.2) + (3 \times 0.3) + (4 \times 0.4) = 0.1 + 0.4 + 0.9 + 1.6 = 3$.

Step 3: Calculation
$E(X^2) = (1^2 \times 0.1) + (2^2 \times 0.2) + (3^2 \times 0.3) + (4^2 \times 0.4) = 0.1 + 0.8 + 2.7 + 6.4 = 10$.
$V(X) = 10 - 3^2 = 10 - 9 = 1$.
Standard Deviation $= \sqrt{V(X)} = 1$.

Step 4: Conclusion
Hence, the values are 3 and 1. Final Answer: (B)