Question

Given the PMF: \(P(X=x) = \alpha\) for \(x = 1,2\), \(= \beta\) for \(x = 4,5\), and \(= 0.3\) for \(x = 3\), with mean \(\mu = 4.2\). Find \(\sigma^2 + \mu^2\)

• A. 20.4
• B. 10.8
• C. 16.4
• D. 21.4

Hint:

Use mean and total probability conditions to find missing probabilities, then compute expected square.

Answer 1

The Correct Option is A

\[ P(1) = P(2) = \alpha, P(3) = 0.3, P(4) = P(5) = \beta \Rightarrow 2\alpha + 0.3 + 2\beta = 1 \Rightarrow \alpha + \beta = 0.35 \] Mean = \(1\cdot \alpha + 2\cdot \alpha + 3\cdot 0.3 + 4\cdot \beta + 5\cdot \beta = 3\alpha + 0.9 + 9\beta = 4.2\)
Solve this with previous: obtain \(\alpha, \beta\), then compute \(E(X^2)\), hence: \[ \sigma^2 + \mu^2 = E(X^2) \Rightarrow \text{Answer} = 20.4 \]