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Statistics Question on Probability

Let 𝑋1, 𝑋2, 𝑋3 be 𝑖. 𝑖. 𝑑. random variables, each having the 𝑁(0, 1) distribution. Then, which of the following statements is/are TRUE?

A

2(X1βˆ’X2)(X1+X2)2+2X32βˆΌπ‘‘1\frac{\sqrt2(X_1-X_2)}{\sqrt{(X_1+X_2)^2+2X^2_3}} βˆΌπ‘‘_1

B

(X1βˆ’X2)2(X1+X2)2+2X32∼F1,2\frac{(X_1-X_2)^2}{\sqrt{(X_1+X_2)^2+2X^2_3}} ∼F_{1,2}

C

E(X1X22+X32)=0E(\frac{X_1}{X^2_2+X^2_3})=0

D

𝑃(𝑋1 < 𝑋2 + 𝑋3 ) =13\frac{1}{3}

Answer

E(X1X22+X32)=0E(\frac{X_1}{X^2_2+X^2_3})=0

Explanation

Solution

The correct option is (C): E(X1X22+X32)=0E(\frac{X_1}{X^2_2+X^2_3})=0