# Binomial distribution problem.

• SithV
In summary, the mode of a binomial B(n, p) distribution is usually equal to ⌊(n+1)p⌋, but 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. The mode can also be expressed as follows: mode = ⌊(n+1)p⌋ if (n+1)p is 0 or a noninteger, mode = (n+1)p and (n+1)p - 1 if (n+1)p is in the range of 1 to n, and mode = n if (n+1
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!

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

SithV said:
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 $\theta$.

## 1. What is the binomial distribution?

The binomial distribution is a probability distribution that describes the likelihood of obtaining a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant throughout the trials.

## 2. How is the binomial distribution different from other probability distributions?

The binomial distribution is unique in that it is discrete, meaning that the outcomes can only take on whole number values, and it is specifically used for a fixed number of trials with two possible outcomes. Other probability distributions, such as the normal distribution, can have a continuous range of values and can be used for a variety of situations.

## 3. What are the parameters of the binomial distribution?

The binomial distribution has two parameters: n, the number of trials, and p, the probability of success in each trial. These parameters are used to calculate the probabilities of different outcomes in the distribution.

## 4. How is the binomial distribution used in real-world applications?

The binomial distribution is commonly used in fields such as finance, biology, and psychology to model situations where there are only two possible outcomes, such as success or failure, yes or no, or heads or tails. It can be used to analyze data and make predictions about future outcomes.

## 5. What is the formula for calculating probabilities in the binomial distribution?

The formula for calculating probabilities in the binomial distribution is P(x) = nCx * px * qn-x, where n is the number of trials, x is the number of successes, p is the probability of success, and q is the probability of failure (1-p). This formula is used to find the probability of getting exactly x successes in n trials.

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