Question 1

A man goes to Atlantic City and as part of the trip he receives a special

20$ coupon. This coupon is special in the sense that only the gains may

be withdrawn from the slot machine. So the man decides to play a game

of guessing red/black. For simplicity let us assume that every time he

plays one dollar. The probability of winning is 1/2. Each time he plays

the 1$ game he has the same strategy. If he loses 1$ is subtracted from

the current coupon amount. If he wins the coupon value remains the

same and he immediately withdraws 1$ to count toward the total gains.

He plays the game until there are no more funds on the coupon. Find

the expected amount of his gains and the expected number of times he

plays the 1$ game.

Question 2

Let {Xi}i>=1 be i.i.d. random variables. Assume that the sums Sn = (Summation from i=1 to n) of Xi

have the property Sn/n approaches 0 almost surely as n approaches infinity.

Showthat E[|X1|] < infinity and therefore E[X1] = 0.

Question 3

The king of Probabilonia has sentenced a criminal to the following

punishment. A box initially contains 999,999 black balls and one white

ball. On the day of sentencing, the criminal draws a ball at random. If

the ball is white, the punishment is over and the criminal goes free. If

the ball is black then two things happen:

(i) the criminal is forced to eat a live toad, and

(ii) the black ball drawn is painted white and returned to the box.

This process is repeated on successive days until the criminal finally

draws a white ball. Let X be the number of toads eaten before the

punishment ends.

(a) Write down a formula which gives P{X = k} exactly.

(b) Estimate the median of X to within three signicant digits.

Question 4

A robot arm solders a component on a motherboard. The arm has

small tiny errors when locating the correct place on the board. This

exercise tries to determine the magnitude of the error so that we know

the physical limitations for the size of the component connections. Let

us say that the right place to be soldered is the origin (0,0), and the

actual location the arm goes to is (X,Y ). We assume that the errors

X and Y are independent and have the normal distribution with mean

0 and a certain standard deviation sigma.

(a) What is the density function of the distance

D = SQRT ( X^2 + Y^2)

(b) Calculate its expected value and variance:

E(D) and Var(D)

(c) Calculate

E[|X^2 - Y^2|]

Question 5

You want to design an experiment where you simulate bacteria living

in a certain medium. To this end you know that the lifetime of one

bacteria is a random variable X (in hours) distributed with exponen-

tial density (1/2)e^(-x/2)

However, you also know that all of these peculiar

bacteria live at least 1 hour and die after 10 hours. Thus you need

to restrict the generated numbers to the interval (1,10) by using a

conditional density.

(a) Give the exact distribution (or density function) for a random

variable which you may use to generate such numbers?

(b) Use any method and write code in any programming language

that allows to generate random numbers with this particular con-

ditional density.

(c) Now, suppose in addition that each of the bacteria individual when

it dies (and only then) it either divides and creates two new in-

dividuals with probability 1/2 or it just dies without any descen-

dants with probability 1/2. Create a program using the lifetime

in the previous part that will keep track of the individuals living

at any moment in time.