Question

Which of the following statements is/are CORRECT? A. If random variables X and Y are independent, then f subscript Y vertical line x end subscript left parenthesis y right parenthesis equals f subscript Y left parenthesis y right parenthesis must hold. B. Given a legitimate f subscript X vertical line y end subscript left parenthesis x right parenthesis , integral subscript negative infinity end subscript superscript plus infinity end superscript f subscript X vertical line y end subscript left parenthesis x right parenthesis d x equals 1 should always holds. C. If sigma subscript X Y end subscript equals 0 then X and Y must be independent. D. P(X >= x) = P( X> x) always holds for continuous random variable X regardless of what value x is.

Answer 1

A) Correct

since if X and Y are independent, then

so using definition of conditional probability we get

B) Correct

As is conditional probability density function so

c) Incorrect

As implies that X and Y are Uncorrelated. And Uncorrelated variables need not to be Independent.

d) Correct

As probability at a point for continuous random variable is always zero.