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Probability most probable value

  1. Sep 7, 2014 #1
    1. The problem statement, all variables and given/known data

    Let ##X## be a discrete random variable, we say that ##x_0 \in R_X## is a most probable value for ##X## if

    ##p_X(x_0)=sup_{x \in R_X} p_X(x)##.

    1)Show that every discrete random variable admits at least one most probable value.

    2) Check that ##[(n+1)p]## is a most probable value for ##X \sim Bi(n,p)##, and that ##[\lambda]## is a most probable value for the Poisson distribution of parameter ##\lambda##


    3. The attempt at a solution

    I am pretty lost in both parts of the problem. As for 2), I know that if ##X## has a binomial distribution ##B(n,p)##, then the mass function

    ##p_X(x)=\binom{n}{x}(1-p)^{n-x}p^x##,

    and that if ##X## has a poisson distribution, then the mass function is

    ##p_X(x)=\dfrac{\lambda^x}{x!}e^{-\lambda}##.

    How can I find the supreme of both functions? Any help would be appreciated.
     
  2. jcsd
  3. Sep 7, 2014 #2

    Ray Vickson

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    Homework Helper

    If
    [tex] b(k) \equiv b_{n,p}(k) = {n \choose k} p^k (1-p)^{n-k},[/tex]
    what is a simple expression for
    [tex] r(k) = \frac{b(k+1)}{b(k)}\:?[/tex]
    So, how can you tell if ##b## is increasing at ##k## (that is, if ##b(k+1) \geq b(k)##)?

    Do something similar for the Poisson distribution.
     
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