# Probability Examples Sheet 3

1. A random variable X has mean μ and variance
For each real number t, let

V (t) = E(X - t)^2. Prove that EV (X) =

2. At time 0, a blood culture starts with one red cell. At the end of one
minute, the red cell

dies and is replaced by one of the following combinations with the following
probabilities.

two red cells (probability 1/4), one red and one white cell (probability 2/3),
two white

cells (probability 1/12). Each red cell lives for one minute and gives birth to
o spring in

the same way as the parent cell. Each white cell lives for one minute and dies
without

reproducing. Individual cells behave independently.

(a) When the culture has been going for just over n minutes, what is the
probability

that no white cells have yet appeared?

(b) What is the probability that the entire culture eventually dies out?

3. A slot machine operates in such a way that at the first turn your
probability of winning

is 1/2. Thereafter, your probability of winning is 1/2 if you lost at the last
turn and p

(which is less than 1/2) if you won. If u_{n} is the probability that you win at the
nth turn,

find a recurrence relation that connects u_{n} and u_{n-1} whenever n≥2. Define a value
for

u_{0} so that this recurrence relation is still valid when n = 1. By solving the
recurrence

relation, prove that

4. A gambler plays the following game. He starts with r
pounds, and is trying to end up

with a pounds. At each go he chooses an integer s between 1 and the minimum of r
and

a - r and then tosses a fair coin. If the coin comes up heads, then he wins s
pounds, and

if it comes up tails then he loses s pounds. The game finishes if he runs out of
money (in

which case he loses) or reaches a pounds (in which case he wins). Prove that
whatever

strategy the gambler adopts (that is, however he chooses each stake based on
what has

happened up to that point), the probability that the game finishes is 1 and the
probability

that the gambler wins is r/a.

5. A fair coin is tossed n times. Let u_{n} be the
probability that the sequence of tosses

never has two consecutive heads. Show that
Find u_{n}, and check

that your value of u_{3} is correct.

6. A coin is repeatedly tossed, and at each toss comes up
heads with probability p, the

outcomes being independent. What is the expected number of tosses until the end
of the

rst run of k heads in a row?

7. Let u_{n} be the number of walks of length 2n
that start and end at the origin, move a

distance 1 at each step, and remain non-negative at all times. (We interpret u_{0}
as 1.) By

considering the last time that such a walk visits the origin before time n,
prove that

Let G(z) be the generating function
Prove that this sum converges whenever

By using the recurrence above, prove also
that Solve this

quadratic to obtain a formula for G(z) (explaining carefully your choice of
sign). Calculate

the first few terms of the binomial expansion of your answer and check that they
give the

right first few values of u_{n}.

8. Let X be a random variable with density f and let g be
an increasing function such

that Find a formula for the density of the
random variable g(X).

If this density is h, check that g(x)f(x) dx.

9. Let be independent
exponential random variables with parameter λ.

Let Prove that Y is Poisson with parameter
λ.

10. Alice and Bob agree to meet at the Copper Kettle after
their Saturday lectures. They

arrive at times that are independent and uniformly distributed between midday
and 1pm.

Each is prepared to wait 10 minutes before leaving. Find the probability that
they meet.

11. The radius of a circle has the exponential
distribution with parameter λ. Determine

the probability density function of the area of the circle.

12. Suppose that X and Y are independent, identically
distributed random variables, each

uniformly distributed on [0, 1]. Let U = X+Y and V = X/Y . Are U and V
independent?

13. Let be a branching
process such that

be the p.g.f. of be the total number of
individuals

in the generations 0, 1, 2, . . . , n, and let
be its generating function. Prove

that Deduce that if
then
satisfies the

equation G(z) = zF(G(z) when 0 ≤ z < 1. (Here we interpret
as 0.) If m < 1, prove

that

14. Let k be a positive integer and let
(0, 1). Find a formula for
. Find also

a formula for