Let Ω = {0, 1} 3 , that is, all possible (ordered) triples of zeros and ones. Suppose that all outcomes have equal probability. We define three random variables X1, X2, and X3 on this space representing the first, second, and third digit, respectively. We also define X = X1 + X2 + X3.
compute: E(E(X|X1)|X2)
Let Ω = {0, 1} 3 , that is, all possible (ordered) triples of zeros and...
Conditional expectations Let 2 - 0, 1)3, that is, all possible (ordered) triples of zeros and ones. Suppose that all outcomes have equal probability. We define three random variables Xi, X2, and X3 on this space representing the first, second, and third digit, respectively. We also define (i) Compute the values (across S2) of each of the following random variables: (i) What is the probability mass function of E(X2 X)
2) Consider the sample space of three coin tosses: Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. Assuming all elements to be equally likely, we assign P({ωi}) = 1/8, i = 1, 2, 3, 4, 5, 6, 7, 8. Define random variable to capture the second and third outcomes of the toss: X2 = { 0, if second outcome is T; 1, if second outcome is H and X3 = { 0, if third outcome is T;...
Let X1,X2 and X3 be three discrete random variables withP[X1 = 0] = P[X1 = 1] = P[X2 = 0] = P[X2 = 1] = 1/2and P[X3 = 0] = 1.(i) Characterize all possible coupling between X1 and X2.(ii) Which coupling maximizes the correlation? Which coupling minimizes thecorrelation? Do you have an intuitive explanation why these couplings are theones that minimize/maximize the correlation?(iii) Which coupling makes the two random variables uncorrelated?(iv) Do the tasks (i) − (iii) but for X1...
QUESTION 15 Let X be a nonnegative random variable (the possible values of X are all nonnegative numbers), and suppose E( X ) = 1, then, the probability that X takes a value greater than 5, cannot be A. larger than 0.1. B. larger than 0.2. C. less than 0.2. D. none of the above. QUESTION 16 Let X be any random variable, and E( X ) = 2, then, the probability that X takes a value greater than 10, cannot...
Let S be the set of distinct ordered triples comprised of the numbers 1, 2, 3, 4. To say that the triple is distinct means that no number occurs twice in the triple. To say that the triple is ordered means that two triples in which the same numbers appear in a different order are considered to be different triples. Some of the elements of S are: 1,2,3), (1,2,4), (3,2,1), (3,2,4), (4,2,1), (4,3,2) We wish to list all of the...
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
Let the mutually independent random variables X1, X2, and X3 be N(0, 1),N(2, 4), and N(−1, 1), respectively. Compute the probability that exactly two ofthese three variables are less than zero.
3. Let {X1, X2, X3, X4} be independent, identically distributed random variables with p.d.f. f(0) = 2. o if 0<x< 1 else Find EY] where Y = min{X1, X2, X3, X4}.
(1) Consider the probability space 2 [0, 1. We define the probability of an event A Ω to be its length, we define a sequence random variables as follows: When n is odd Xn (u) 0 otherwise while, when n is even otherwise (a) Compute the PMF and CDF of each Xn (b) Deduce that X converge in distribution (c) Show that for any n and any random variable X : Ω R. (d) Deduce that Xn does not converge...
2. Let S be the sample space of a single toss of a fair coin. Define the sequence of random variables X, on S as follows: (I Ifs-T (a) Are X1.x2 . Convergent almost surely? (b) Find P((s E S : limx,(s)-1)).
2. Let S be the sample space of a single toss of a fair coin. Define the sequence of random variables X, on S as follows: (I Ifs-T (a) Are X1.x2 . Convergent almost surely? (b) Find P((s...