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 Let X and Y denote the number of red and blue balls that are withdrawn.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