Homework #1. Due January 30 |

1.Associate with the space of infinite (countable) number of coin tossings a set X of (Bernoulli) sequences x= (x

a. Define an elementary event (interval) I

b. Assume that the (probability) measure m of an elementary event is 1/2 and that for two different k,n elementary events I

m(I

Hint: Consider an event (1,1,1,1,1,1,....).

c. Construct the sample space of possible events F - the (minimal) sigma-algebra that contains all elementary events and the outer (probability) measure on this sigma-algebra. Are the sets of events F and F

2. Probability of long leads. The Arc-sine law.

Suppose a population of n people casts their votes for Al or Bill at random with probability 1/2. This can be viewed as a coin tossing model with a random variable (measurable function) g

a. Show that g

b. We say that Al is winning at time t=n if g

A

is a measurable event.

c. Using Matlab (or other software) and performing numerical simulations compute numerically the distribution of the function

f

d. Compare (numerically) probabilities that

"Al is winning in less than .02 n out of n tossings" with the probability "Al is winning in larger .49 n but less than .51 n out of n tossings".

"Al is winning in larger than .02 n out of n tossings" with the probability "Al is winning in larger .49 n but less than .51 n out of n tossings".

"The most fortunate contender is winning in larger than .976 n out of n tossings" with the probability "The most fortunate contender is winning in less than .524 n out of n tossings".

What is your conclusion?

e. Let f = lim

Note. f

The explanation of the arc-sine law lies in the fact that frequently enormously many trials are required before the particle returns to the origin. Geometrically speaking, the path crosses the time axis very rarely. We feel intuitively that if Al and Bill toss a coin for a long time 2n, the number of ties should be roughly proportional to 2n. This is not so. Actually the number of ties increases in probability only as (2n)

f. Using Matlab (or other software) and performing numerical simulations compute numerically the distribution of the function f

g. Let f = lim

One should be warned that the number of returns is not normally distributed.

The next three exercises are manifestations of the Littlewood's three principles

3. Prove the following theorem(see hints for this exercise in Royden problem 13 p. 64)

Let E be a given subset of R (or R

4. Egorov's theorem. (see Royden problem 30 p. 74)

If < f

5. Lusin's theorem.(see Royden problem 31 p. 74)

Let f(x) be a measurable real-valued function on [a,b], and let > >0 be arbitrary. Then there is a continuous function g(x) such that the measure of the set {x:| f(x) g(x)} is less than .

6. Continuity of the Lebesgue integral (see also Royden problem 5 p. 89)

Let f(x) be an integrable function. Show that the function F(x) defined by

F(x) =

is a continuous function.

7. Convergence theorems for "convergence in measure" (see Royden problem 21 p. 96)

Prove that Fatou's lemma and the Monotone and Lebesgue Convergence theorems remain valid if "convergence a.e." is replaced by "convergence in measure".

8. Green's function for the Poisson equation on a unit disk D in R

For the Poisson equation

u(x) = f(x), x=(x

u(x) = -1/(4 )

The function G(x)=-1/(4 |x|) is called the Green's function for the Poisson equation on a unit disk D in R

a. Suppose f(x) =1. Then u(x) = -1/(4 )

Suggestion: you can use Mathematica (or other software) to compute it.

Answer: u(x) = |x|

b. Define a discrete Laplacian

c. Show that almost everywhere G(x-y) = 0 for any fixed y. Hence

d. Using integral convergence theorems explain the "contradiction" for u(x) defined in part a. u(x) =1, but