1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Variance of binomial distribution - 1 trial

  1. Jul 31, 2012 #1
    1. The problem statement, all variables and given/known data

    For n trials, [itex]S_n[/itex] can be seen as the sum of n independent single trials [itex]X_i[/itex], i = 1,2,...,n, with [itex]\mathbb{E}[X_i][/itex]=p and Var[itex][X_i]=p(1-p)[/itex].

    2. What I don't understand

    I don't understand why Var[itex][X_i]=p(1-p)[/itex].

    We know that: Var[itex][X_i]=\mathbb{E}[(X_i - \mathbb{E}[X_i])^2] = \mathbb{E}[X_i^2 - 2X_i\mathbb{E}[X_i] + \mathbb{E}[X_i]^2] = \mathbb{E}[X_i^2] - \mathbb{E}[X_i]^2[/itex].

    Taking [itex]\mathbb{E}[X_i]^2[/itex], we have [itex]\mathbb{E}[X_i]^2=p^2[/itex].

    Taking [itex]\mathbb{E}[X_i^2][/itex], we have [itex]\mathbb{E}[X_i^2]=\mathbb{E}[\prod_{i=1}^2X_i] = \prod_{i=1}^2 \mathbb{E}[X_i] = \mathbb{E}[X_i]^2 = p^2[/itex].

    So Var[itex][X_i]= p^2 - p^2 = 0 \not= p(1-p)[/itex], which contradicts what my lecture notes say.
  2. jcsd
  3. Aug 1, 2012 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No, [itex] E X_i^2 \neq p^2.[/itex] In fact, [itex]E f(X_i) = \sum_{k} P\{X_i = k\} f(k)[/itex] for any function f, so we get [itex] E X_i^2 = p.[/itex]

  4. Aug 1, 2012 #3
    Enlightening post, thank you.

    I guess my error was assuming that X_i was independent in:
    [itex]\mathbb{E}[X_i^2]=\mathbb{E}[\prod_{i=1}^2X_i] = \prod_{i=1}^2 \mathbb{E}[X_i] = \mathbb{E}[X_i]^2 = p^2[/itex].


    But they're the same event so that would be non-sensical....
  5. Aug 1, 2012 #4
    Use the definition of variance of a discrete variable.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Variance of binomial distribution - 1 trial
  1. Binomial Distribution (Replies: 4)

  2. Binomial Distribution (Replies: 1)

  3. Binomial distribution (Replies: 1)

  4. Binomial distribution (Replies: 2)