Question

Let 𝑋1 and 𝑋2 be 𝑖. 𝑖. 𝑑. random variables having the common probability density function
\(f(x) =\begin{cases}   e^{-x}     & \quad x ≥0,\\  0, & \quad Otherwise \end{cases}\)
Define 𝑋₍₁₎=min{𝑋₁, 𝑋₂} and 𝑋₍₂₎ = max{𝑋₁, 𝑋₂}. Then, which one of the following statements is FALSE?

• A. \(\frac{2x_{(1)}}{X_{(2)}-X_9(1)}\)~ F2, 2
• B. \(2(X_{(2)}-X_{(1)})\)~ \(X^2_2\)
• C. \(E(X_{(1)})=\frac{1}{2}\)
• D. 𝑃(3𝑋₍₁₎< 𝑋₍₂₎)=\(\frac{1}{3}\)

Answer 1

The Correct Option is D

To solve the problem, we first need to understand the properties of the random variables. The given common probability density function (pdf) is:

\(f(x) = \begin{cases} e^{-x}, & \quad x \geq 0, \\ 0, & \quad \text{otherwise} \end{cases}\) 

This is an exponential distribution with parameter \(\lambda = 1\).

Step 1: Analyzing the Order Statistics

Given two independent and identically distributed (i.i.d.) random variables \(X_1\) and \(X_2\):

• \(X_{(1)} = \min(X_1, X_2)\) is the first order statistic (smallest value).
• \(X_{(2)} = \max(X_1, X_2)\) is the second order statistic (largest value).

Step 2: Verify Statements

Let's verify each of the statements provided in the options to determine which one is FALSE:

Option A: \(\frac{2X_{(1)}}{X_{(2)}-X_{(1)}} \sim F_{2,2}\)

Indeed, for exponential i.i.d. random variables, it is known that:

\(\frac{2X_{(1)}}{X_{(2)}-X_{(1)}} \sim F_{2,2}\)

This statement is TRUE.

Option B: \(2(X_{(2)}-X_{(1)}) \sim \chi^2_2\)

This statement is TRUE for exponential random variables.

Option C: \(E(X_{(1)}) = \frac{1}{2}\)

For an exponential distribution with two i.i.d. variables:

\(E(X_{(1)}) = \frac{1}{2}\)

This statement is TRUE.

Option D: \(P(3X_{(1)} < X_{(2)}) = \frac{1}{3}\)

To analyze this, consider that for exponential variables:

\(X_{(1)} \sim \text{Exponential}(2)\), and \(X_{(2)} - X_{(1)} \sim \text{Exponential}(1)\)

The probability calculation for this does not equate to \(\frac{1}{3}\), hence this statement is FALSE.

Conclusion

The FALSE statement is indeed Option D: \(P(3X_{(1)} < X_{(2)}) = \frac{1}{3}\)