Question

Let (𝑋₁, 𝑋 ) be a random vector having the probability mass function

where 𝑐 is a real constant. Then, which of the following statements is/are TRUE?

• A. 𝐸(𝑋₁ + 𝑋₂ )=8
• B. \(𝑉𝑎𝑟(𝑋_1 + 𝑋_2 ) =\frac{8}{3}\)
• C. \(𝐶𝑜𝑣(𝑋_1,𝑋_2 ) = −\frac{5}{9}\)
• D. 𝑉𝑎𝑟(𝑋₁ + 2𝑋₂)=8

Answer 1

The Correct Option is A, B, D

We are given a random vector \( (X_1, X_2) \) with the probability mass function:

The probability mass function is defined as:

\[ f(x_1, x_2) = \begin{cases} \frac{c}{x_1! x_2! (12 - x_1 - x_2)!}, & \text{if } x_1, x_2 \in \{0, 1, \dots, 12\}, x_1 + x_2 \leq 12 \\ 0, & \text{otherwise.} \end{cases} \]

To verify the correct statements, consider the following:

1. Expectation: \[ E(X_1 + X_2) = 12 \times \frac{2}{3} = 8 \] This matches with statement: 𝐸(𝑋₁ + 𝑋₂) = 8.
2. Variance of \( X_1 + X_2 \):
   • We calculate the variance using: \[ Var(X_1 + X_2) = Var(X_1) + Var(X_2) + 2Cov(X_1, X_2) \] Given the condition and symmetry, it simplifies to: \[ Var(X_1 + X_2) = \frac{8}{3} \]
   This confirms the statement: 𝑉𝑎𝑟(𝑋₁ + 𝑋₂ ) = \(\frac{8}{3}\).
3. Covariance:
   • The statement regarding covariance \( Cov(X_1, X_2 ) = −\frac{5}{9} \) is not verified based on the given information.
4. Variance of \( X_1 + 2X_2 \):
   • Calculate using the formula: \[ Var(X_1 + 2X_2) = Var(X_1) + 4Var(X_2) + 4Cov(X_1, X_2) \] From the conditions, results in: \[ Var(X_1 + 2X_2) = 8 \]
   This confirms the statement: 𝑉𝑎𝑟(𝑋₁ + 2𝑋₂) = 8.

Thus, the correct statements are:

• \(𝐸(𝑋_1 + 𝑋_2) = 8\)
• \(𝑉𝑎𝑟(𝑋_1 + 𝑋_2 ) = \frac{8}{3}\)
• \(𝑉𝑎𝑟(𝑋_1 + 2𝑋_2) = 8\)