[SOLVED] Math 425 Fall 2024 – HW 11 Prolog

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 ].