Math 425 Fall 2024 - HW 11
Due Friday 11/15, 11:59pm, via Gradescope
Please note:
(1). Please include detailed steps. Only providing the result will not get full credits.
(2). Please write at most one problem in each page. If you reach the bottom please start a new page instead of writing two columns in one page. If a problem contains multiple small questions, you may write them in one page.
(3). Please associate pages with problems in Gradescope.
1. Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius 1 centered at the origin. Find the joint density function fR,Θ(r, θ) of the polar coordinates R =
√X2 + Y2 and Θ = arctan(X/Y).
2. If X1 and X2 are independent exponential random variables, both having parameter λ, find the joint density function fY1,Y2
(y1, y2) with Y1 = X1 + X2 and Y2 = e
X1.
3. Amy throws a fair die and simultaneously flips a fair coin. If the coin lands head, then she wins twice of the value that appears on the die. If tails, then she wins one-half of the die value. Determine her expected winnings.
4. If X and Y have joint density function
(1). Find E[XY ].
(2). Find E[X].
(3). Find E[Y ].