Question

We take a random sample X1,X2,…X35 from a distribution with mean μ. Consider the following estimators:

Θ^1=X11.

Θ^2=(X11+X31+X33+X35)4

Θ^3=(2X33+4X35)5

a) Compute the following expectations:

(Enter your answers in terms of μ)

E(Θ^1)=

E(Θ^2)=

E(Θ^3)=

b) Hence, which of Θ^1, Θ^2 and Θ^3 are unbiased for μ?

Θ^1

Θ^2

Θ^3

Answer 1

The expected values of the three estimators are: E(Θ^1) = X11/35, E(Θ^2) = X11/35, and E(Θ^3) = (2X33 + 4X35)/35. Θ^1 and Θ^2 are unbiased for μ, while Θ^3 is not.

Step-by-step explanation:

To find the expected value, E(X), or mean μ of a discrete random variable X, you need to multiply each value of the random variable by its probability and add the products. The formula is given as E(X) = µ = Σ xP(x).

a) For Θ^1, E(Θ^1) = E(X11) = X11 * P(X11) = X11 * (1/35) = X11/35. This is unbiased for μ.

For Θ^2, E(Θ^2) = E(X11 + X31 + X33 + X35)/4 = (X11+X31+X33+X35)/4 * P(X11+X31+X33+X35). Since X11, X31, X33, and X35 are independently distributed, the expectation of their sum is a sum of their expectations. So, E(Θ^2) = 4 * E(X11)/4 = E(X11) = X11/35. This is also unbiased for μ.

For Θ^3, E(Θ^3) = E(2X33 + 4X35)^5 = (2X33 + 4X35) * P(2X33 + 4X35) = (2X33 + 4X35)/35. This is not unbiased for μ.

b) Θ^1 and Θ^2 are unbiased estimators for μ, while Θ^3 is not.