Answer:
a. ½(Y² - Y) is an unbiased estimator of the variance.
b. 2 Standard Error = 2√(Y²/2 - Y/2)
Explanation:
Given
E(Y) = 1/p
V(Y) = (1 - p)/p²
Simplifying V(Y)...
V(Y) = 1/p² - 1/p
Because V(Y) = 1/p² - 1/p and E(Y) = 1/p,
We'll guess that there might be an unbiased estimator of shape
aY² + bY.
Solving E(aY² + bY)...
First, E(Y²) = V(Y) + (E(Y))²
Substituting the values of V(Y) and E(Y);
E(Y²) = 1/p² - 1/p + (1/p)²
E(Y²) = 1/p² - 1/p + 1/p²
E(Y²) = 2/p² - 1/p
With the above,
E(aY² + bY) is then equal to
E(aY² + bY) = 2a/p² - a/p + b/p
The above equation is equal to V(Y), if and only if
a = ½ and b = -½
-------------- Checking------------
Let E(aY² + bY) = V(Y)
i.e.
2a/p² - a/p + b/p = 1/p² - 1/p
Multiply through by l
2a/p - a + b = 1/p - 1
Comparing right hand side to left hand side
2a/p = 1/p ----- Equation 1
And
- a + b = - 1 ------- Equation 2
Solving Equation 1 (Multiply both sides by p)
2a = 1
So, a = ½
Substitute ½ for a in Equation 2
-½ + b = -1
b = -1 + ½
b = -½
------------ End --------------
Thus ½(Y² - Y) is an unbiased estimator of the variance.
b.
2 Standard Error is given by
2√V(Y)
= 2√(1/p² - 1/p)
= 2√(Y²/2 - Y/2)
