Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

INTRODUCTORY/PRINCIPLES OF MATHEMATICAL STATISTICS (STAT2001/6039/2013/6013)

Semester 1, 2022

Assignment 1

2pts   (a)    Explain why 匝Xn  = 0 for any positive odd integer n. Hint: use properties of evenfunctions.

1pts   (b)    How about 匝X 一 1 ? Is that also zero? Why or why not?

8pts   (c)    Find 匝┌│Z│┐, 匝 ┌│X│┐, 匝 ┌│Z│3 ┐and 匝Œ│X│3 ┐ .

1pts   (d)    Explain why 匝ŒZeZ2 =2] is not zero even though h.z/ = zez2 =2 is an even function.

Problem 2    An experiment is repeated independently (at times 1, 2, 3,. . . ) where in each experiment, one ball is chosen, red with probability p, green with probability q and blue with probability 1 - p - q (all three probabilities being between 0 and 1.)

Let X be the first time that red is chosen and Y the first time that green is chosen.

1pts   (a)    What are the marginal distributions of X and Y ?

3pts   (b)    Write down the joint probability mass function p.x; y/ of X and Y . Be sure to specify the set of valid points .x; y/. Hint: you should probably write the x > y and x < y cases separately.

2pts   (c)    Hence write down the conditional probability mass function of X given Y = y .

2pts   (d)    What is the conditional expectation of X given that Y = y?

Problem 3

2pts    (a)    If X ～ Gamma.˛; ˇ/, find 匝 ┌ X s .aX ┐, specifying the values ofs and a (with a always positive)

for which it exists.

2pts   (b)   Suppose that K is a random variable that only takes values in {0; 1; 2; 3; : : :}. Show that

&

匝ŒK] =      贮{K > k}:

k =0

4pts   (c)   Suppose that we start with three red balls and one green ball in a bag. On each turn, a ball is drawn uniformly at random from the bag. If the ball is green, it is returned to the bag together with an extra green ball, and play continues. If the ball drawn is red, the game ends. Let K be the number of draws up to and including the ending (red) draw. Find a simple expression in terms of k for the probability that K  > k.  Hence (using (b)) or otherwise, calculate the expected value of K. Does K have a finite second moment (i.e., finite variance)?