Suppose that the random variable Y1,...,Yn satisfy Yi = ?xi + ?i i=1,...,n. where the set...

Suppose that the random variable Y₁,...,Y_n satisfy Y_i = ?x_{i +} ?_i i=1,...,n. where the set of x_i are fixed constants and ?_i are iid random variables following a normal distributions of mean zero and variance ?^2.

?_a (with a hat on it) = ?_i=1ⁿY_i x_i/ ?_i=1ⁿx²_i is unbiased estimator for ?.

The variance is ?_a (with a hat on it) = ?²/ ?_i=1ⁿx²_i . What is the distribation of this variance?

Expert Solution

orchestra answered 2 years ago

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

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The observations are Y1, . . . , Yn. The model is Yi = βxi +...

The observations are Y1, . . . , Yn. The model is Yi = βxi + i , i = 1, . . . , n, where (i) x1, . . . , xn are known constants, and (ii) 1, . . . , n are iid N(0, σ2 ). Find the MLEs of β and σ^ 2 . Are they jointly sufficient for β and σ ^2 ?

Suppose Xi and Yi are all independent (i=1,2,3), where the three Xi are iid and follow...

Suppose Xi and Yi are all independent (i=1,2,3), where the three Xi are iid and follow an Exponential distribution with rate r, while the three Yi are also iid but follow a Normal(μ, σ2) distribution. (a) Write down the joint pdf for the random vector (X1,X2,X3,Y1,Y2,Y3). (b) Find the expected value of the product X1 Y1, i.e., E(X1 Y1), and find Cov(X2, X3).

Suppose Xi and Yi are all independent (i=1,2,3), where the three Xi are iid and follow...

Suppose Xi and Yi are all independent (i=1,2,3), where the three Xi are iid and follow an Exponential distribution with rate r, while the three Yi are also iid but follow a Normal(µ, σ2) distribution. (a) Write down the joint pdf for the random vector (X1,X2,X3,Y1,Y2,Y3). (b) Find the expected value of the product X1 Y1, i.e., E(X1 Y1), and find Cov(X2, X3).

Let D = {xi} n i=1, where xi ∈ R, be a data set drawn independently...

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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)

Let Y1, ... , Yn be a random sample that follows normal distribution N(μ,2σ^2) i)get the...

Let Y1, ... , Yn be a random sample that follows normal distribution N(μ,2σ^2) i)get the mle for σ^2 ii)prove using i) that it is an efficient estimator

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

Suppose Y1, . . . , Yn are independent random variables with common density fY(y) = eμ−y y > μ. Derive a 95% confidence interval for μ. Find the MLE for μ to use as the estimator

We have a dataset with n = 10 pairs of observations (xi; yi), and Xn i=1...

We have a dataset with n = 10 pairs of observations (xi; yi), and Xn i=1 xi = 683; Xn i=1 yi = 813; Xn i=1 x2i = 47; 405; Xn i=1 xiyi = 56; 089; Xn i=1 y2 i = 66; 731: What is the line of best t for this data?

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Suppose that the random variable Y1,...,Yn satisfy Yi = ?xi + ?i i=1,...,n. where the set...

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Expert Solution

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