# summation

• Oct 8th 2008, 03:50 PM
lord12
summation
1. Show that Var[X-E[X]] = Var[X]. Var[X] = E[X^2]-(E[X])^2
2. Ten Hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability 0<p<1, compute the expected number of duck that escape unhurt when a flock of size 10 flies overhead.
• Oct 8th 2008, 04:13 PM
mr fantastic
Quote:

Originally Posted by lord12
1. Show that Var[X-E[X]] = Var[X]. Var[X] = E[X^2]-(E[X])^2
[snip]

1. Since E(X) is a number, you want to show that $\displaystyle Var(X - \mu) = Var(X)$. And since $\displaystyle Var(aX + b) = a^2 Var(X)$, ......

Proofs of $\displaystyle Var(aX + b) = a^2 Var(X)$ and $\displaystyle Var(X) = E(X^2)-[E(X)]^2$ can be found in almosy any stats textbook. And there are probably numerous websites that give proofs. So I'm not going to re-invent the wheel .....
• Oct 8th 2008, 04:19 PM
tukeywilliams
2. Look at it from the perspective of a duck. The probability of a duck being shot is $\displaystyle 1/n$. This is a binomial probability, since the trials are independent. Let $\displaystyle h$ be the number of hunters and $\displaystyle n$ be the number of ducks.

So the expected number of hunters shooting at a particular duck is: $\displaystyle h/n$.

Then the probability that no one will shoot at that duck is: $\displaystyle \left(1-\frac{1}{n}\right)^{h} = \left(\frac{n-1}{n} \right)^{h}$.

And so the expected number of ducks with nobody shooting them is $\displaystyle n \cdot \left(\frac{n-1}{n} \right)^{h}$. Here $\displaystyle n=h=10$. This is $\displaystyle \approx 3.48$ ducks. Probably round down to $\displaystyle 3$ for it to make sense in the context of the problem (conservative guess). Or round up to $\displaystyle 4$.