Suppose Y1, . . . , Yn are independent random variables with common density fY(y) =...

Suppose Y₁, . . . , Y_n are independent random variables with common density f_Y(y) = e^μ−y y > μ.

Derive a 95% confidence interval for μ. Find the MLE for μ to use as the estimator

Expert Solution

We will use info above

orchestra answered 2 years ago

Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent random samples, with...

Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent random samples, with the variables Xi normally distributed with mean μ1 and variance σ12 and the variables Yi normally distributed with mean μ2 and variance σ22. The difference between the sample means, X − Y, is then a linear combination of m + n normally distributed random variables and, by this theorem, is itself normally distributed. (a) Find E(X − Y). (b) Find V(X − Y). (c)...

Suppose that Y1 ,Y2 ,...,Yn is a random sample from distribution Uniform[0,2]. Let Y(n) and Y(1)...

Suppose that Y1 ,Y2 ,...,Yn is a random sample from distribution Uniform[0,2]. Let Y(n) and Y(1) be the order statistics. (a) Find E(Y(1)) (b) Find the density of (Y(n) − 1)2 (c) Find the density of Y(n) − Y (1)

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y...

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y | X = x is modeled by Y=β0+β1x+ε, ε∼N(0,σ^2), where β0,β1and σ^2 are unknown. Let β1 denote the mle of β1. Derive V(βhat1).

2 .Define the probability density functions (pdfs) of the independent random variables X and Y as:...

2 .Define the probability density functions (pdfs) of the independent random variables X and Y as: fX(x)={ (1/5).(1-(x/10)), 0≤x≤10 fY(y)={ (1/5).(y/10) , 0≤y≤10 0 , otherwise for both fX(x) and fY(y) (a) Suppose that you are playing a game where you want to maximize your number of points , and you can have your score be X or Y . Which would you choose? (b) Suppose I am confused in part (a), so I flip an unfair coin, with P...

Suppose Y1,Y2, .. ,Y8 are independent and identically distributed as Poisson random variables with mean lambda....

Suppose Y1,Y2, .. ,Y8 are independent and identically distributed as Poisson random variables with mean lambda. a) Derive the most powerful test for testing Ho: lambda = 2, Ha: lambda = 3. Carefully show all work involved in the derivation. i) Give the form of the test. (In other words, for what general values of Y1,Y2, .. ,Y8 will Ho be rejected?) ii) Describe the rejection region as carefully as possible if alpha <= .05 (and is as close as...

Let Y1,...,Yn have common pmf, p(y)= 2p if y = -1, 1/2 - p if y...

Let Y1,...,Yn have common pmf, p(y)= 2p if y = -1, 1/2 - p if y = 0, 1/2 - p if y = 1. a.) Find a sufficient statistics for p. b.) Find the MOM estimator of p. Is it unbiased? c.) Find the MLE estimator of p. Is it unbiased? d.) Find the MSE of both estimators. Using MSE as a criterion, which estimator is better?

Let Y1,...,Yn be a sample from the Uniform density on [0,2θ]. Show that θ = max(Y1,...

Let Y1,...,Yn be a sample from the Uniform density on [0,2θ]. Show that θ = max(Y1, . . . , Yn) is a sufficient statistic for θ. Find a MVUE (Minimal Variance Unbiased Estimator) for θ.

Suppose that X1 and X2 are independent continuous random variables with the same probability density function...

Suppose that X1 and X2 are independent continuous random variables with the same probability density function as: f(x)= x/2 0 < x < 2, 0 otherwise. Let a new random variable be Y = min(X1, X2, ). a) Use distribution function method to find the probability density function of Y, fY (y). b) Compute P(Y > 1). c) Compute E(Y ).

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2,...

1. Suppose that Z1,Z2 are independent standard normal random variables. Let Y1 = Z1 − 2Z2, Y2 = Z1 − Z2. (a) Find the joint pdf fY1,Y2(y1,y2). Don’t use the change of variables theorem – all of that work has already been done for you. Instead, evaluate the matrices Σ and Σ−1, then multiply the necessary matrices and vectors to obtain a formula for fY1,Y2(y1,y2) containing no matrices and no vectors. (b) Find the marginal pdf fY2 (y2). Don’t use...

Suppose X and Y are independent random variables with X = 2:8 and Y = 3:7....

Suppose X and Y are independent random variables with X = 2:8 and Y = 3:7. Find X+Y , the standard deviation of X + Y .

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