# Binomial distribution problem.

1. Nov 6, 2011

### SithV

If X is a binom. rand. var., for what value of θ is the probability b(x;n,θ) at max?
Ive no idea...
My only guess (most likely wrong) is that max and min are always derivatives...
So do i just differentiate and express θ...?
Any suggestions...?=(
Thank you!

2. Nov 7, 2011

### mathman

From Wikipedia

Usually the mode of a binomial B(n, p) distribution is equal to ⌊(n + 1)p⌋, where ⌊ ⌋ is the floor function. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows:

\text{mode} = \begin{cases} \lfloor (n+1)\,p\rfloor & \text{if }(n+1)p\text{ is 0 or a noninteger}, \\ (n+1)\,p\ \text{ and }\ (n+1)\,p - 1 &\text{if }(n+1)p\in\{1,\dots,n\}, \\ n & \text{if }(n+1)p = n + 1. \end{cases}

I

3. Nov 7, 2011

### Stephen Tashi

That statement doesn't make sense. What you might mean is that you guess that this problem involves taking the derivative of a function and finding what values of the variable make it zero, in order to find the function's max or min. Yes, that is correct.

Do you know what function to differentiate?

Remember in max-min problems, if the variable is restricted to an interval you also have to check the endpoints of the interval as well as finding where the derivative is zero. Since $\theta$ is a probability, it is restricted by $0 \leq \theta \leq 1$.

Those remarks are relevant to maximizing $B(x,n,\theta)$ with respect to $n$. If the original post states the problem correctly, it is to maximize $B(x,n,\theta)$ with respect to $\theta$.