Math 425 Fall 2024 - HW 12
Due Friday 11/22, 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.
For density functions we omit the statement ”f=0 otherwise” for con-venience.
1. Consider N independent flips of a coin having probability p of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, for HHT HT there are 3 changeovers. Find the expected number of changeovers.
Hint: Express the number of changeovers as the sum of Bernoulli random vari-ables.
2. The joint density function of X and Y is
Find E[X] and E[Y], and show that Cov(X, Y) = 1.
3. Use the same density function as in Problem 2 to find E[X2|Y = y].
4. A fair 6-side die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a ”6” and a ”5”.
(1). Find E[X].
(2). Find E[X|Y = 1].
(3). Find E[X|Y = 5].
5. The joint density function of X and Y is
Find E[Y3|X = x].
6. An urn contains 30 balls: 10 red, 8 blue, 12 yellow. Pick 12 balls randomly. Let X and Y denote the number of red and blue balls that are withdrawn. Calculate Cov(X, Y) by defining appropriate indicator (Bernoulli) random vari-ables