[Solved] MA2631 Assignment 7

1. (continuing problem 2, Assignment 6): Consider the random variable X with the probability mass distribution

P[X = 1] = 0.3, P[X = 4] = 0.25, P[X = 7] = 0.4, P[X = 10] = 0.05.

Calculate the variance of X and Y with Y = 3X + 2.

2. Suppose we pick a month at random from a non leap-year calendar and let X be the number of days in that month. Find the mean and the variance of X.
3. Let Y be a binomial distributed random variable with n trials of success probability p. Show that

Var[Y ] = np(1 p).

4. Let Z be a geometric distributed random variable with success probability p. Calculate Var[Z].
5. Assume that X is a random variable taking values on the non-negative integers that satisfies

P[X n + i|X n] = P[X i].

Show that X is a geometric distributed random variable.

2

6. The number of errors on a book page follow a Poisson distribution. It has been determined that on 10% of the pages there is at least one error.
   1. Determine the parameter of the Poisson distribution.
   2. What is the expected number of errors on a page?

8 points per problems

Standard Carlton and Devore, Section 2.3: 32, 39 44; Section 2.4: 50, 54, 59, 61, 66, Section 2.5: 76, 81; Section 2.6: 98, 99, 102

Extra Random Walk: Consider the whole numbers and start at 0. You consecutively flip a coin, and if it shows heads you move one to the right (+1) while for tails you move one to the left (1). This models a random movement, e.g., approximately the behavior of a (very) drunk person. Denote by X_n the random variable that marks your position after n steps.

• What are expectation and variance of X_n? What happens if you consider for them the limit n ?
• After k steps, you are at some point x. How likely is it that you will return to the starting point zero before time n? How does this probability behave for n ?