O T O Statistics and Probability 11. Let 1.2.... be a sequence of iid zero-mean random variables. Let (a,) be a sequence of positive constants such that E a, = 1. Define the sequence of random variables Xo, X1, X2, ... via the recursion Xo = (0, X, = (1 - a,)X-1 + a, En n= 1, 2,... (a) Show that the collection {4;} is uniformly integrable. (b) Use proposition 3.36 to show that {X,} is uniformly integrable. Hint: For a convex function f : R - R and a e (0, 1), f(ax + (1- a)y) < af(x) + (1 - a)f(y).

Suppose that X;; i = 1,2, 3, .. n are independent random variables with a common uniform distribution over (0, 1) and Yg is distributed over (0, Yg -1) with Y, =1,k = 1,2, ..., n. Let Z = k = 1,2,3, .n R.D.R EGALA STAT 203 March 4, 2021 Show by mathematical induction induction or any other method that Zg has probability density function (-In z)*-1 fR(z) = k = 1,2,3, . n (k – 1)! Hence deduce that Zg and Y, have the same distribution (k = 1,2,3, ..n)

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.

For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete.   (d) X = the length of a randomly selected rattlesnake Describe the set of possible values for the variable. 0, 1, 2, 3, ... 1, 2, 3, ...     [0, ∞) (0, ∞) State whether the variable is discrete. discrete not discrete     (e) Z = the sales tax percentage for a randomly selected amazon.com purchase, where n = the total number of possible sales tax percentages for online purchases Describe the set of possible values for the variable. z1, z2, z3, ..., zn [0, zn] 1, 2, 3, ..., n [0, n] State whether the variable is discrete. discrete not discrete     (f) Y = the pH of a randomly chosen soil sample Describe the set of possible values for the variable. 0, 1, 2, 3, ..., 14 [0, 14] 1, 2, 3, ..., 14 (0, 14] State whether the variable is discrete. discrete not discrete     (g) X = the tension (psi) at which a randomly selected…