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.

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

Let rt be a log return. Suppose that r0, r1, . . . are i.i.d. N(0, 0.01^2).(a) What is the distribution of rt(8) = rt + rt−1 + rt−2 +...+ rt−7?(b) What is the covariance between r7(3) and r9(3)?(c) What is the conditional distribution r17(3) given that r16 =0.004 (d) What is the probability that the gross return over the first 10 times periods is at least 1.05?

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.

List all possible simple random samples of size n = 2 that can be selected from the population {0, 1, 2, 3, 4}. Calculate V(x) for the sample. Q2 (a) A dentist interested in the effectiveness of a new toothpaste. A group of N= 1000 school children participated in this study. Pre-study records showed that there is an average of 2.2 cavities every six months for the group. After three months of the study, the dentist sampled n= 10 children to determine how they are progressing on the new toothpaste. The data can be shown in Table Q2(b). Construct a 95% confidence interval for x. (b) Table Q2(b) Number of cavities in the Children three-month period 1 4 3 2 4 5 2 6. 7 3 8 4 1 10 1 The Fish and Game Department of a state was concerned about the direction of its future hunting programs. To provide for a greater potential for future hunting, the department wanted to determine the proportion of hunters seeking any type of game bird. A simple random sample of n= 1000 of the N= 99,000 licensed…

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