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!

Homework Help: Binomial Distribution Probability

  1. Sep 29, 2009 #1
    Let X be a Binomial B([tex]\frac{1}{2}[/tex],n), where n=2m.

    Let a(m,k) = [tex]\frac{4^m}{(\stackrel{2m}{m})}P(X = m + k)[/tex].

    Show that [tex]lim_{m->\infty}(a(m,k))^2 = e^{-k^2}[/tex].

    So far, I've found that P(X = m+k) = [tex](\stackrel{2m}{m+k}) \frac{1}{4^m}[/tex]

    Then, a(m,k)=[tex]\frac{m!m!}{(m+k)!(m-k)!}.[/tex]

    But I have no idea how to show that the limit of [tex]a^2[/tex] will be equal to [tex]e^{-k^2}[/tex].

    I think my work up to that point seems okay. I got things to cancel out, so that is usually a good sign. Any hints? Thank you!
  2. jcsd
  3. Sep 29, 2009 #2
    I got what you got, and this is not a good sign. If we are correct, then look at the k=3 case.

    a(m,3) simplifies to [tex]\frac{m(m-1)(m-2)}{(m+3)(m+2)(m+1)}[/tex] and the limit is 1 as m approaches infinity.
  4. Sep 29, 2009 #3
    Well, we are looking for the limit of [tex]a^2[/tex]. Does that make a difference?

    Also, you said you got the same thing as me, but I'm a little confused how what you wrote (even though I know it's for the specific k=3 case) is the same as what I got. I'll look at it again to see if I can see the connection, but maybe what I got for a was wrong?
  5. Sep 30, 2009 #4
    [tex]a(m,3)=\frac{m!m!}{(m+3)!(m-3)!}=\frac{m!\cdot m(m-1)(m-2)(m-3)!}{(m+3)(m+2)(m+1)m!\cdot (m-3)!}



    I think you should check if the problem is stated correctly. I get the same thing as you did for a(m,k) every time I check.
  6. Sep 30, 2009 #5
    Yep, that's definitely what the problem says so, I'm not sure.

    What you've shown definitely seems right and I'm assuming it works for other cases when k is something else, as well. I'll check it out and ask my teacher about it if need be. Thanks!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook