The number of persons coming through a blood bank until the first person with type A blood is a random variable Y with a geometric distribution. If p denotes the probability that any one randomly selected person will possess type A blood, then E[Y ] = 1/p and V [Y ] = (1-p)/p2.
Find a function of Y that is an unbiased estimator of V [Y ].
Then suggest how to form a two-standard error bound on the error of estimation when Y is used to estimate 1/p.

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)

a manufacturer of printed circuit boards uses exponential smoothing with trend to forecast monthly demand of its product. at the end of december, the company wishes to forecast sales for january. the estimate of trend through november has been 200 additional boards sold per month. average sales have been around 1000 units per month. the demand for december was 1100 units. the company uses α = 0.20 and β = 0.10. make a forecast including trend for the month of january.

answerboth the short run phillips curve and the aggregate demand and aggregate supply model