For what value of θ is the binomial probability b(x;n,θ) maximized?

[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!

 

[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}

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[Stephen Tashi]

If X is a binom. rand. var., for what value of θ is the probability b(x;n,θ) at max?

My only guess (most likely wrong) is that max and min are always derivatives...

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.

So do i just differentiate and express θ...?

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 $0 \leq \theta \leq 1$.

Usually the mode of a binomial B(n, p) distribution is ...

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 $\theta$.