Part CCSIR NET June 2025the-survival-function-of-s-applied-to-s-is-uniform-and-a-chi-square-has-variance-2k

The survival function of s applied to s is uniform and a chi square has variance 2k

Let X and Y be independent and identically distributed N(0, 1) random variables. Let $S = X^{2} + Y^{2}$ and T = $e^{- (X^{2} + Y^{2}) /2}$ . Then, which of the following statements are true?

A.The probability density function of S is f_S(s) = $(1/2) e^{- s /2}$ if s > 0, and 0 otherwise.✓
B.The probability density function of T is f_T(t) = 1 if 0 < t < 1, and 0 otherwise.✓
C.Var(S) = 2.
D.E(T) = 2/3.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

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The survival function of s applied to s is uniform and a chi square has variance 2k

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