QUESTION 5 Let X,Y be joint Gaussian random variables with E[X]=0,E[Y]=0, Var[X]=2, Var[Y]=4 and Cov[X,Y]=2. Compute Var[XIY=1] as a decimal and enter the answer.
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![QUESTION 5
Let X, Y be joint Gaussian random variables with E[X]=0,E[Y]=0, Var[X]=2, Var[Y]=4 and Cov[X,Y)=2. Compute
Var[XIY=1] as a decimal and enter the answer.
Transcribed Image Text: QUESTION 4 Let X, Y be joint Gaussian random variables with E[X]=0,E[Y]=0, Var[X]-2, Var[Y]=4 and
Cov[X,Y] =2. Compute E[XIY=1] as a decimal and enter the answer. QUESTION 5 Let X, Y be joint Gaussian random variables with
E[X]=0,E[Y]=0, Var[X]=2, Var[Y]=4 and Cov[X,Y] =2. Compute Var[X]Y=1] as a decimal and enter the answer.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?Question 3 A product is classified according to the number of defects it contains and the factory that produces it. Let X and Y be the random variables that represent the number of defects per unit (taking on possible values of 0, 1, 2, or 3) and the factory number (taking on possible values 1 or 2), respectively. The entries in the table represent the joint possibility mass function of a randomly chosen product.
- 2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.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-Y2Let X and Y be random variables, and a and b be constants. a) Prove that Cov(aX, bY) = ab Cov(X,Y). b) Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.
- Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0Let X and Y be random variables, and a and b be constants. ???? a) Show that Cov [aX,bY] = abCov [X,Y] . b) Show that if a > 0 and b > 0, then the correlation coefficient between aX and bY is the same as the correlation coefficient between X and Y . c) Is the correlation coefficient between X and Y unaffected by changes in the units of X and Y ?f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.
- 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!)Let Xi be IID random variables which have the same law as X. Let L(t) = E(e^tX.) Suppose that this is well defined for t ∈ [−1, 1]. Express the moment generating function of the Sum from i=1 to k Xi in terms of k and LLet 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.