Question

The probability distribution of a random variable X is given by

Then the variance of X is

• A. 1.76
• B. 2.45
• C. 1.56
• D. 4.8

Hint:

$Var(X) = E(X^2) - \mu^2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the variance of a discrete random variable given its probability distribution table.
Step 2: Key Formula or Approach:
1. \( E(X) = \mu = \sum x_i P(x_i) \)
2. \( E(X^2) = \sum x_i^2 P(x_i) \)
3. \( \text{Var}(X) = E(X^2) - [E(X)]^2 \)
Step 3: Detailed Explanation:
From the table: \( X_i = \{0, 1, 2, 3, 4\} \), \( P_i = \{0.4, 0.3, 0.1, 0.1, 0.1\} \).
Calculate \( E(X) \):
\[ E(X) = 0(0.4) + 1(0.3) + 2(0.1) + 3(0.1) + 4(0.1) = 0 + 0.3 + 0.2 + 0.3 + 0.4 = 1.2 \] Calculate \( E(X^2) \):
\[ E(X^2) = 0^2(0.4) + 1^2(0.3) + 2^2(0.1) + 3^2(0.1) + 4^2(0.1) = 0 + 0.3 + 0.4 + 0.9 + 1.6 = 3.2 \] Calculate \( \text{Var}(X) \):
\[ \text{Var}(X) = 3.2 - (1.2)^2 = 3.2 - 1.44 = 1.76 \] Step 4: Final Answer:
The variance is 1.76.