Let Y1, Y2, . . . , Y20 be a random sample of size n =...

Let Y1, Y2, . . . , Y20 be a random sample of size n = 20 from a normal distribution with unknown mean µ and known variance σ 2 = 5. We want to test H0; µ = 7 vs. Ha : µ > 7. (a) Find the uniformly most powerful test with significance level 0.05. (b) For the test in (a), find the power at each of the following alternative values of µ: µa = 7.5, 8.0, 8.5, and 9.0. (c) Sketch a graph of the power function (d) What is the smallest sample size n such that an α = 0.05-level test has power at least 0.8 when µ = 8?

Expert Solution

orchestra answered 3 years ago

Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample...

Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample of size n = 4 from a distribution with pdf f(x) = 3X2, 0 < x < 1, zero elsewhere. (a) Find the joint pdf of Y3 and Y4. (b) Find the conditional pdf of Y3, given Y4 = y4. (c) Evaluate E(Y3|y4)

Let Y1, Y2, . . ., Yn be a random sample from a uniform distribution on...

Let Y1, Y2, . . ., Yn be a random sample from a uniform distribution on the interval (θ - 2, θ). a) Show that Ȳ is a biased estimator of θ. Calculate the bias. b) Calculate MSE( Ȳ). c) Find an unbiased estimator of θ. d) What is the mean square error of your unbiased estimator? e) Is your unbiased estimator a consistent estimator of θ?

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

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Let Y1, Y2, ..., Yn be a random sample from an exponential distribution with mean theta....

Let Y1, Y2, ..., Yn be a random sample from an exponential distribution with mean theta. We would like to test H0: theta = 3 against Ha: theta = 5 based on this random sample. (a) Find the form of the most powerful rejection region. (b) Suppose n = 12. Find the MP rejection region of level 0.1. (c) Is the rejection region in (b) the uniformly most powerful rejection region of level 0.1 for testing H0: theta = 3...

Let y1, y2, .....y10 be a random sample from an exponential pdf with unknown parameter λ....

Let y1, y2, .....y10 be a random sample from an exponential pdf with unknown parameter λ. Find the form of the GLRT for H0: λ=λ0 versus H1:λ ≠ λ0. What integral would have to be evaluated to determine the critical value if α were equal to 0.05?

For any nonnegative integer n, let Y1 < Y2 < · · · < Y2n+1 be...

For any nonnegative integer n, let Y1 < Y2 < · · · < Y2n+1 be the ordered statistics of 2n + 1 independent observations from the uniform distribution on [−2, 2]. (i) Find the p.d.f. for Y1 and the Y2n+1. (ii) Calculate E(Yn+1). Use your intuition to justify your answer.

Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a)...

Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1 0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1, Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).

Let Y1,Y2,...,Yn be a Bernoulli distributed random sample with P(Yi = 1) = p and P(Yi...

Let Y1,Y2,...,Yn be a Bernoulli distributed random sample with P(Yi = 1) = p and P(Yi = 0) = 1−p for all i. (a) Prove that E(¯ Y ) = p and V (¯ Y ) = p(1−p)/n2, for the sample mean ¯ Y of Y1,Y2,...,Yn, and find a suﬃcient statistic U for p and show it is suﬃcient for p. (b) Find MVUE for p and show it is unbiased for p.

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...

Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y = (Y1,Y2,Y3)′ using : Multivariate normal distribution properties.

Suppose that Y1 and Y2 are random variables with joint pdf given by f(y1,y2) = ky1y2...

Suppose that Y1 and Y2 are random variables with joint pdf given by f(y1,y2) = ky1y2 ; 0 < y1 <y2 <1, where k is a constant equal to 8. a) Find the conditional expected value and variance of Y1 given Y2=y2. b) Are Y1 and Y2 independent? Justify your answer. c) Find the covariance and correlation between Y1 and Y2. d) Find the expected value and variance of Y1+Y2.

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