We consider a sequence (X^)^-1 of independent random variables such that P(X=1)=1- P(Xn = −1) = Pn. n For all n < N, let So = 0 and S₁ == > X₂. We also consider the filtration (F)-1 given by i=1 Fn=σ(X1,..., Xn). (d) Is there a choice of the values of pn (n> 1) such that (S2) 0 is a martingale ? (e) Consider the process (V)-1 defined by n ΔΥΠ Vn == = ASn Prove that (n)=1 is previsible. Yn - Yn-1 Sn - Sn-1

Question 2 Let X be a random variable with moment generating function M(t), and let Y= ax+b. Prove that the moment generating function of Y has this form: M(b+at) e^(bt)xM(at) e^(bt)+M(at) e^(at)xM(bt)

Suppose that the rate of increase of a bighorn ram's horn is approximated by 0.42 x² - 3.97 x + 31.24 cm per year, where x is between 2 and 5 years. (These numbers have been randomized from the actual equation given in the project done in discussion section.) Find the total change in the length of a bighorn ram's horn between the ages of 2 and 5. Give the answer as a decimal number with at least two decimal places.

A random walk (RW) {Sn}n>o is a sum of id r.v.s X1, X2, •• , Xn,withP(X1 = a) = p, P(X1 = b) = 1 -p= q.Find the expectation E(Snl and variance var(Sn) of Sn for any n.

If we let X1, X2,...X10 be a random sample from the distribution N( μ, σ), then what is the value k so that: P( (X-μ)/(σ/√10) < k ) = .05 P( (X-μ)/(S/√10) < k ) = .05 P( k < (X-μ)/(S/√10) ) = .05

Suppose the distribution of some random variable X is modeled as Negative binomial with parameters, p and r, where p is the probability of success and r is the total number of success over the trials. Find the method of moment (MM) estimators for (p,r), in closed algebraic form expressions based on a random sample of size n, X1, X2,...,Xn.

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

If P(B) = ² and P(A'^B) = 12, for two events A and B, find P(A/B).

Question 1.2 Consider the function  f (x) = (1/24(x^2 +1)      1 < or =  x < or = 4)          = (0      otherwise)   Calculate P (x = 3) Calculate P (2 < or =  x < or = 3)         Question 1.3 Consider the function  f (x) = (k - x/4      1 < or =  x < or = 3)          = (0      otherwise)   which is being used as a probability density function for a continuous random variable x?   a. Find the value of K b. Find P (x < or = 2.5)

The joint PDF of two random variables X,Y is given byfXY(x, y) = {k e^(-y-x/4) , x > 0, y > 0                    0,                  otherwisea)  What is the value of k ?b)  Find the probability P( X<Y ).c)  Find P( 0<X<2 ).