Given a marginal p.m.f of x (f₁(x)) that distribute (EJ)uniformly on 1 and 2and a marginal p.m.f of y (f₂(y)) that distribute (EJ)uniformly on 10 and 22x and y are stochastically independent r.v.sthen the J.p.m.f of x and y (f(x,y) ) isfor x = 1,2f(x, y) =y = 10,22,O.WOption 2f(x, y) =0f(x, y) =for x = 1,2y = 10,22,O.WOption 1(1 for x = 1,2y = 10,22,O.W0-{²0

Answered: Given a marginal p.m.f of x (f₁(x))…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Prove the following
2. Suppose that x1, x2,..., In is a random sample drawn from a population whose density function is distributed as Exponential (VA). Suppose two possible esti- mators of A as 1 n ÂA = and AB= 2n i=1 1 n(n + 1) 2 (Σ=)² f ==— e J N (a) Are the two estimators are unbiased? Then, compute the MSE of each estimator. (b) Can you make an optimality statement about any of the proposed estima- tors Based on CRLB. What happens if n→ ∞o?
2. Suppose that Y1 ~ Exp(3) and Y2 ~ Exp(2) are independent. (a) Determine the joint pdf of Y1 and Y2. (b) Use the Jacobian method to determine the pdf of U = Y1 + Y2. (c) Use the mgf method to determine the distribution of V = 4Y1 + 6Y2. The distribution of V will be one of the "famous" ones in your Super Handout.
b) Let X₁, X₂, ..., Xn and Y₁, Y₂, ..., Ym be random samples from populations with moment generating functions Mx₁(t) = e³t+t² and My(t) = (₁)², respectively. iii) What is the value of the sample size m, if P(|Ỹ − µỹ| ≥ 10) < 0.04?
Consider a (non)-symmetric random walk M = Xo +X1+X2+…+Xn where X;'s (i> 1) are independent and identical distributed, P(X; = 1) = p and P(X;=-1) =1–p and Xo = 0. Let Sn = eaS,-n Denote Fn= 0(X1,...,Xn). Find a such that (Sn)n20 is a (Fn)n20 - martingale.
4. X(t)=Acos(wt+q)=f(q,t), where A and are constants, is a random variable uniformly distributed in the interval [0; л/2]. Find if the random process X(t) is ergodic with respect to mean value. Find if the random process X(t) is ergodic with respect to variance and covariance.
4. Suppose Y₁, Y2,..., Yn are iid from uniform(1,0). That is f(y; 0) = ₁,1 < y < 0. Find a sufficient statistic for 0.
b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?
20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2
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.
Let X, Y, Z be the three assets, with VAR(X) = VAR(Y) = VAR(Z) = 1. Z is independent from X and Y , and COV(X, Y) = 0.2 .Assuming no short selling, what is the portfolio weight on Z that gives the smallest variance? (Hint: since Z is independent with X and Y, you may express wx and wy as a function of 1- wz , and then find wz.)
2. Assuming the dichotomous disease model, with Y1 and Y2 being two relatives with relatedness R, show that (a) Cov(Y1, Y2) = P(Y1 = Y2 = 1) – K². (b) express R as a function of the covariance and K.

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