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Binomial density

  1. Sep 9, 2010 #1
    hello,

    i need to prive that for a binomial r.v X E[X]=NP and VAR(X)=NP(1-P).

    I tried to prove it using the deffinition of expectation:

    E[x]=[tex]\sum xi \stackrel{N}{i} p^{i}(1-p)^{n-i}[/tex]

    now what?

    thanks...
     
  2. jcsd
  3. Sep 9, 2010 #2

    mathman

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    Science Advisor
    Gold Member

    E[x]=[tex]\sum xi \stackrel{N}{i} p^{i}(1-p)^{n-i}[/tex] is incorrect. Should be:
    E[x]=[tex]\sum i \stackrel{N}{i} p^{i}(1-p)^{n-i}[/tex]

    For the second moment replace i by i2.
     
  4. Sep 11, 2010 #3

    statdad

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    Homework Helper

    One more comment: to help with the variance, instead of calculating

    [tex]
    E[X] = \sum i^2 \binom{N}{i} p^i (1-p)^{n-i}
    [/tex]

    calculating

    [tex]
    E[X (X-1)] = \sum i(i-1) \binom{N}{i} p^i (1-p)^{n-i}
    [/tex]

    will make the algebra required to work with the summation simpler.

    Since

    [tex]
    E[X(X-1)] = E[X^2] - E[X]
    [/tex]

    this, and the mean, will allow you to find the variance.
     
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