Part CCSIR NET December 2023shifted-cauchy-sign

Shifted cauchy sign

Suppose X is a continuous random variable with density $f (x) = (1/ π) \cdot 1/ (1 + (x + 1)^{2})$ , and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?

A.E(Y) = 0
B.P(Y > 0) < P(Y < 0)
C.P(Y < −1) < P(Y > 1)
D. $E (Y^{2}) = 1$

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The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: Uncorrelated ⇒ independent

Shifted cauchy sign

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