5. Suppose X is a random variable.(a) X is independent of itself if and only if there is some constant c such thatP[X = c] = 1.(b) If there exists a measurableg: (R, B(R)) → (R, B(R)),such that X and g(X) are independent, then prove there exists c ER suchthatP[g(X) = c] = 1.

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Answered: 5. Suppose X is a random variable. (a)…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Let X be a continuous random variable described below: p(X) = x/4, for all x in the range 0 ≤ x ≤ 2 = x2/8, for all x in the range 2 < x ≤ k (a) What is the value of k?(b) What is E(X)? Note: Answers to both questions are irrational numbers.
Let X be a random variable on a closed and bounded interval [a, b] whose CDF is continuous. Let g(x) be a convex function. Prove that g (E (X)) ≤ E(g(X))
Q1 Let X1 and X2 be independent exponential random variables with identical parameter A. Q1(i.) Find the distribution of Z = max(X1, X2). Q1 (ii.) Find the distribution of Y = min(X1, X2). Q1(iii.) Calculate E[Y]. Q1(iv.) Calculate E[Z]. Q1(v.) Using the relation Z = X1+X2 – Y, Calculate E[Z] and verify that it agrees with the calculation done in part (iv.)
A) 0.4625 B) 0.3375 C) 0.600 D) 0.4355
Let W = 2X + 5Y – 3Z, where the random variables X, Y, and Z satisfy E(X)= 2, V(X)= 5, E(Y)= -3, V(Y)=4, E(Z)=7, V(Z) = 6, Cov(X,Y) = -1, Cov(Y, Z) =3, Cov(Z, X)= 2. Calculate E(W), V (W) and Cov(W, Y).
Example 2.1.1 Let S = R, and suppose P is a probability measure on R. Define F(x) by F(x) = P((-∞, x]), XER. (2.3) Then (i) F is right continuous,
Let (X1, X2, X3) be the random outcome of tossing a fair six-faced die three times, and let (Y(1), Y(2), Y(3) be the order statistics from (X1, X2, X3). We have Y(1) = min (X1, X2, X3), Y(3) = max (X1, X2, X3), and Y(1) < Y(2) < Y(3). Find Pr (Y(1) = 3).
Let X1, X2,..., Xn be independent exponential variables, parameter λ. Show by induction that X2 S = X₁ + X₂ + + Xn has the r(λ, n) distribution.
Example 2.19 Let X be a random variable with possible values {x1, x2,...x) such that P(X = x) =!v i. Find E(X).
3. Consider the linear probability model Y₁ = Bo+B₁Xi+ui, where Pr(Y₁ = 1|Xi) = Bo +3₁ Xi. (a) Show that E(u₂|X₂) = 0. (b) Show that Var(u₁|X;) = (Bo + B₁X;)[1 − (Bo + B₁X;)]. (c) Is ui conditionally heteroskedastic? Is u heteroskedastic? (d) Derive the likelihood function.
3. Let the random variable X have the pdf f(x) = 2(1 — x), 0 ≤ x ≤ 1, zero elsewhere. a) Find the cdf of X. Provide F(x) for all real numbers x (set up the appropriate cases). b) Find P(1/4 < X < 3/4). c) Find P(X= 3/4). d) Find P(X ≥ 3/4).
Let X be a random variable on a closed and bounded interval [a, b]. Let g(x) be a convex function. Prove that g(E(X)) ≤ E (g(X)

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5. Suppose X is a random variable. (a) X is independent of itself if and only if there is some constant c such that P[X = c] = 1. (b) If there exists a measurable g: (R, B(R)) → (R, B(R)), such that X and g(X) are independent, then prove there exists c E R such that P[g(X) = c] = 1.