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: Understanding the Question:
We are given a discrete probability distribution and need to calculate the mean ($\mu$) and the standard deviation ($\sigma$).

Step 2: Key Formula or Approach:
1. Mean $\mu = E(X) = \sum x_i P(x_i)$.
2. Variance $\sigma^2 = E(X^2) - [E(X)]^2 = \sum x_i^2 P(x_i) - \mu^2$.
3. Standard Deviation $\sigma = \sqrt{\text{Variance}}$.

Step 3: Detailed Explanation:
First, calculate the mean $E(X)$:
\[ E(X) = 1(0.1) + 2(0.2) + 3(0.3) + 4(0.4) \] \[ E(X) = 0.1 + 0.4 + 0.9 + 1.6 = 3.0 \] Next, calculate $E(X^2)$:
\[ E(X^2) = 1^2(0.1) + 2^2(0.2) + 3^2(0.3) + 4^2(0.4) \] \[ E(X^2) = 0.1 + 4(0.2) + 9(0.3) + 16(0.4) \] \[ E(X^2) = 0.1 + 0.8 + 2.7 + 6.4 = 10.0 \] Now, calculate the variance:
\[ \sigma^2 = E(X^2) - [E(X)]^2 = 10 - 3^2 = 10 - 9 = 1 \] The standard deviation is:
\[ \sigma = \sqrt{1} = 1 \]

Step 4: Final Answer:
The mean is 3 and the standard deviation is 1.