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Statistics Question on Univariate Distributions

Let 𝑋1 and 𝑋2 be 𝑖. 𝑖. 𝑑. random variables having the common probability density function
f(x)={e−xx≥0, 0,Otherwisef(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

2x(1)X(2)−X9(1)\frac{2x_{(1)}}{X_{(2)}-X_9(1)}~ F2, 2

B

2(X(2)−X(1))2(X_{(2)}-X_{(1)})~ X22X^2_2

C

E(X(1))=12E(X_{(1)})=\frac{1}{2}

D

𝑃(3𝑋(1)< 𝑋(2))=13\frac{1}{3}

Answer

𝑃(3𝑋(1)< 𝑋(2))=13\frac{1}{3}

Explanation

Solution

The correct option is (D): 𝑃(3𝑋(1)< 𝑋(2))=13\frac{1}{3}