Probability most probable value

[mahler1]

Homework Statement

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 $\lambda$

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.

 

[Ray Vickson]

Homework Statement

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 $\lambda$

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.

If
$$b(k) \equiv b_{n,p}(k) = {n \choose k} p^k (1-p)^{n-k},$$
what is a simple expression for
$$r(k) = \frac{b(k+1)}{b(k)}\:?$$
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.