Exercise 13. Let X1,..., Xn ~N (μx, σ²) i.i.d. and Y₁,..., Yn ~N(μy, σ²) i.i.d. with known σ².We assume that for each i the random variables X and Y are correlated, with Corr(Xi, Yi) = 1/2.Define Di = Xi - Y₁ for all i = {1,..., n}. Finally, let X, Y and D be the averages of the Xi, Yiand Di, respectively.a) Show that Cov(Xi, Yi) = 0²/2.b) Determine E(D) and Var(Di).c) Consider the test for Ho: px = μy which uses the test statisticZ = √NDand which rejects Ho, if and only if |Z|> 1.96. Show that P(type I error)= 5%.d) In two or three sentences, discuss the differences between the test from part (c) on the onehand, and the non-paired test for comparing the means of two populations on the other

Hi! Thank you for the question. As you have posted multiple sub-parts, according to our policy, we…

Answered: 2 Exercise 13. Let X1,..., Xn ~N(x, 02)…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

Consider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)
Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?
Let X1 and X2 be two independent random variables. Suppose each Xi is exponentially distributed with parameter λi. Let Y=Min (X1, X2). A) Find the pdf of Y. B) Find E(Y). Hint: Let Y = Min (X1, X2). 1. P[Y > c] = P[Min (X1, X2) > c] = P[X1 > c, X2 > c] 2. Obtain the pdf of Y by differentiating its cdf of Y.
Let Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50
Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.
Let U1, ....U5 be independent and standard uniform distibuted random variables given by P(U1 ≤ x) = x, 0 < x < 1 1. Compute the moment generating function E(e sU ) of the random variable U1. 2. Compute the moment generating function of the random variable Y = aU1 + U2 + U3 + U4 + U5 with a > 0 unknown. 3. Compute E(Y ) and V ar(Y ). 4. As an estimator for the unknow value θ = a we migth use as an estimator θb = 2 n Xn i=1 Yi − 4 = 2Y − 4. with Yi independent and identically distributed having the same cdf as the random variable Y discussed in part 2. Compute E(θb) and V ar(θb) and explain why this estimator is sometimes not very useful. 5.Give an upperbound on the probability P(| θb− a |> ) for every > 0.(Hint:Use Chebyshevs inequality!)
Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…
For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:
X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1=(X1/r1)/(X2/r2) and Y2=X2. (a) Find the joint pdf of Y1 and Y2.
Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.
1)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).

SEE MORE QUESTIONS

Recommended textbooks for you

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

Elementary Statistics: Picturing the World (7th E…

Statistics

ISBN:

9780134683416

Author:

Ron Larson, Betsy Farber

Publisher:

PEARSON

The Basic Practice of Statistics

Statistics

ISBN:

9781319042578

Author:

David S. Moore, William I. Notz, Michael A. Fligner

Publisher:

W. H. Freeman

Introduction to the Practice of Statistics

Statistics

ISBN:

9781319013387

Author:

David S. Moore, George P. McCabe, Bruce A. Craig

Publisher:

W. H. Freeman

SEE MORE TEXTBOOKS