Let the joint distribution of X and Y be given by f(x,y)=begincasescxye^-(x^2+y^2); & x0, y00; & otherwiseendcases With reference to the above, consider the following statements: I. The value of c for f(x,y) to be a proper pdf is 4. II. The probability P(X^22Y^2) is 1/3. III. X and Y are independently distributed. Which of the statements given above are correct?

Q32Section I (Objective (Q1–Q80))32 of 79

Distribution Theory — checking normalization, a probability, and independence

Let the joint distribution of $X$ and $Y$ be given by

$f(x,y)=\begin{cases}cxye^{-(x^2+y^2)}; & x>0, y>0\\0; & \text{otherwise}\end{cases}$

With reference to the above, consider the following statements:

$\text{I. The value of } c \text{ for } f(x,y) \text{ to be a proper pdf is } 4.$

$\text{II. The probability } P(X^2>2Y^2) \text{ is } 1/3.$

$\text{III. } X \text{ and } Y \text{ are independently distributed.}$

Which of the statements given above are correct?

Pick an option to check your answer. In a mock test a wrong answer here costs 0.83 mark.

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