Results 1 to 3 of 3

Math Help - summation

  1. #1
    Junior Member
    Joined
    Aug 2007
    Posts
    71

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by lord12 View Post
    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 Var(X - \mu) = Var(X). And since Var(aX + b) = a^2 Var(X), ......

    Proofs of Var(aX + b) = a^2 Var(X) and 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 .....
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member tukeywilliams's Avatar
    Joined
    Mar 2007
    Posts
    307
    2. Look at it from the perspective of a duck. The probability of a duck being shot is  1/n . This is a binomial probability, since the trials are independent. Let  h be the number of hunters and  n be the number of ducks.

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

    Then the probability that no one will shoot at that duck is:  \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  n \cdot \left(\frac{n-1}{n} \right)^{h} . Here  n=h=10 . This is  \approx 3.48 ducks. Probably round down to  3 for it to make sense in the context of the problem (conservative guess). Or round up to  4 .
    Last edited by tukeywilliams; October 8th 2008 at 04:29 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to do a Summation
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: May 14th 2011, 11:08 AM
  2. Summation
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: May 7th 2011, 06:23 AM
  3. summation help
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 16th 2009, 02:08 PM
  4. summation
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 2nd 2009, 08:17 AM
  5. Summation Help
    Posted in the Algebra Forum
    Replies: 9
    Last Post: January 31st 2009, 07:47 PM

Search Tags


/mathhelpforum @mathhelpforum