We consider a sequence (X)-1 of i.i.d. random variables such that P(X₁ = 1) = P(X₁ = −1) = 1/2. Consider the filtration (F)1 given by = Fn(X1,..., Xn). For all n N, let Sn == Xbe the random walk. We have seen in class that the process (f(Sn)) is a martingale if and only if f satisfies D2f = 0, where n=0 D2f(x) = f(x + 1) − 2f(x) + f(x − 1). - (a) Use the result above to prove again that (en) No is not a martingale. n=0 (b) We consider a process (Y)-0 given by Y₁ = f(n, Sn). Show that (Yn) is a martin- gale provided that - f(n+1, x + 1) − 2f(n, x) + f(n + 1, x − 1) = 0, - for all n, x. (c) Let Y = pen with p = ete. Use (b) to prove that (Y) is a martingale. 2 (d) Since (Y) is a martingale, use the Martingale Representation Theorem guarantees that there exists a previsible process (Zn) such that Y₁ = In = (Z · S)n. Determine Zn and prove that it is previsible. 1

4. If the random variable X has an exponential distrib show that for a > 0, b>0, 1 f(x) = -x/², for x ≥ 0, e 2 P(X> a+b|x> a) = P(X > b).

For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 1.

Thank you.

The random variable x is N(10,1). Find f(x(x-10)²<4).

The moment-generating function of the discrete random variable X is 1 M(t) =e 1 1 +. -2t 2t -3t e+ 6. . 3 What is the value of P(X> 1) ) 6.

Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).

8- 6. Show that, for any random variable, X, and a > 0, 8 心 P(x

3. Let X be a continuous random variable on the interval [0, 1] having cdf Fx(x) = 3x² − 2x³. Let W = 4 + 3X. (a) What is the range of W? (b) Find a formula for Fw(w). (c) Find a formula for fw(w).

67. Let X be a discrete random variable with pmf (a) Find the pmf for Y = X². (b) Find the pmf for U = X + 2. f(x) = px(x) = 1/8, 1/4, 1/4, 1/4, 1/8, 0, x = -2 x = -1 x = 0 x = 1 x = 2 otherwise