Question

Suppose that E[X]= E[Y] = mu, where mu is a fixed unknown number. We have independent simple random samples of size n each from the distribution of X and Y, respectively. Suppose that Var[X] = 2*Var[Y]. Consider the following estimators of mu:

m1 = bar{X}

m2 = bar{Y}/2

m3 = 3*bar{X}/4 + 2*bar{Y}/8

where bar{X} and bar{Y} are the sample mean of X and Y values, respectively. Which of the estimators are unbiased?

Answer #1

**Given that**

**1)
**

**So,**

**So, the first estimator is unbiased.**

**2)
**

**So,**

**So, the second estimator is not unbiased.**

**3)
**

**So,**

**So, the third estimator is unbiased.**

**So, the answer is m1 and m3 are unbiased
estimators.**

Suppose X and Y are independent variables with E(X) = E(Y ) = θ,
Var(X) = 2 and Var(Y ) = 4. The two estimators for θ, W1 = 1/2 X +
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(1) Are W1 and W2 unbiased? (2) Which estimator is more
efficient (smaller variance)?

Suppose that X and Y
are random samples of observations from a population with mean μ
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Consider the following
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A = (2/3)X + (1/3)Y B
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Find variance of A.
Var(A) and Var(B)
Efficient and unbiased
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Consider the following two unbiased point estimators of μ.
A = (7/4)X - (3/4)Y B = (1/3)X +
(2/3)Y
[Give your answers as ratio (eg: as number1 /
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2. Find variance of B. Var(B)
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given x,y are independent random variable. i.e
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Find cov(M2,M1).

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The pdf of W is:
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Poisson
Binomial
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Suppose we have the following values for the linear function
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