Question:
Let (π1, π ) be a random vector having the probability mass function
where π is a real constant. Then, which of the following statements is/are TRUE?
Let (π1, π ) be a random vector having the probability mass function
where π is a real constant. Then, which of the following statements is/are TRUE?
where π is a real constant. Then, which of the following statements is/are TRUE?
Updated On: Nov 17, 2025
- πΈ(π1 + π2 )=8
- \(πππ(π_1 + π_2 ) =\frac{8}{3}\)
- \(πΆππ£(π_1,π_2 ) = β\frac{5}{9}\)
- πππ(π1 + 2π2)=8
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The Correct Option is A, B, D
Solution and Explanation
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:
- Expectation: \[ E(X_1 + X_2) = 12 \times \frac{2}{3} = 8 \] This matches with statement: πΈ(πβ + πβ) = 8.
-
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: \[ Var(X_1 + X_2) = \frac{8}{3} \]
-
Covariance:
- The statement regarding covariance \( Cov(X_1, X_2 ) = β\frac{5}{9} \) is not verified based on the given information.
-
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: \[ Var(X_1 + 2X_2) = 8 \]
Thus, the correct statements are:
- \(πΈ(π_1 + π_2) = 8\)
- \(πππ(π_1 + π_2 ) = \frac{8}{3}\)
- \(πππ(π_1 + 2π_2) = 8\)
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