This homework follows the standard late penalty: 0% if in the stat310 mailbox by Thursday 8 Mar 4pm, 10% by 5pm the following day, 100% otherwise. Please read the syllabus for other homework policies.
(Bonus homework worth half a normal homework)
(5 points) Let \( f(x, y) = c(x + 2xy + 2y) \), \( x \in [0,1], y \in [0,1] \)
What is \( c \) ?
What is \( F(x, y) \)?
(8 points). Which of the following bivariate pdfs represent the pdf of two independent pdfs? (You can assume \( x \in [0,1], y \in [0,1] \) and you don’t need to find the values of any constants). Show your reasoning.
\( f(x, y) = c_1 e^{x+y} \)
\( f(x, y) = c_2 (x + y) \)
\( f(x, y) = c_3 (xy + x + y + 1) \)
\( f(x, y) = c_4 (x^2 y^2 + x^2 y + x^2) \)
(6 points) \( X \sim Unif(0, 10) \), \( Y | X = x \sim Exp(\theta = x) \). Find:
\( f(x, y) \)
\( f(y) \)
\( E(Y) \)