Question

Consider a discrete random variable \( X \) whose probabilities are given below. The standard deviation of the random variable is ......... (round off to one decimal place). 

Hint:

To calculate the standard deviation for a discrete random variable, first calculate the expected value \( E(x) \), then the expected value of \( x^2 \). Subtract the square of the expected value from the expected value of \( x^2 \) to get the variance, and take the square root to find the standard deviation.

Answer 1

Given: \[ x_1 = 1, \, 2, \, 3, \, 4 \quad {and} \quad P(X = x_i) = 0.3, \, 0.1, \, 0.4, \, 0.3 \] Now, the expected value \( E(x) \) is calculated as: \[ E(x) = \sum x_i P(x_i) = 1 \times 0.3 + 2 \times 0.1 + 3 \times 0.4 + 4 \times 0.3 = 4.1 \] Next, the expected value of \( x^2 \) is: \[ E(x^2) = \sum x_i^2 P(x_i) = 1^2 \times 0.3 + 2^2 \times 0.1 + 4^2 \times 0.3 + 4^2 \times 0.3 = 24.7 \] The variance is: \[ V(x) = E(x^2) - (E(x))^2 = 24.7 - (4.1)^2 = 7.89 \] Finally, the standard deviation is: \[ \sigma = \sqrt{7.89} = 2.808 \] Thus, the standard deviation of the random variable is \( 2.8 \).