Thread: Grad school probability concerns

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be withdrawn from the slot machine.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.

I did the first one. I need help on the others please.