Question

Maximum of binomial distribution. Find the value \(n=n^{*}\) that causes the function $$ W=\frac{N !}{n !(N-n) !} p^{n}(1-p)^{N-n} $$ to be at a maximum, for constants \(p\) and \(N\). Use Stirling's approximation, \(x ! \simeq(x / e)^{x}(\) see Appendix B). Note that it is easier to find the value of \(n\) that maximizes \(\ln W\) than the value that maximizes \(W\). The value of \(n^{*}\) will be the same.

Short answer

The value of \(n\) that maximizes \(W\) is \(n^{*} = Np\).

Step-by-step solution

State the problem

To find the value of \(n = n^{*}\) that maximizes the function \(W\). It's easier to find the value that maximizes \(\ln W\), and the value will be the same.

Write the natural logarithm of W

Compute the natural logarithm of \(W\): \(\ln W = \ln \left( \frac{N!}{n!(N-n)!} p^n(1-p)^{N-n} \right)\).

Apply Stirling's Approximation

Using Stirling's approximation \(x! \approx \left(\frac{x}{e}\right)^x\), rewrite \(\ln W\).

Simplify the logarithmic expression

Substitute the approximations: \(\ln W \approx \ln \left( \frac{(N/e)^N}{(n/e)^n ((N-n)/e)^{N-n}} \right) + n \ln p + (N-n) \ln (1-p)\). Simplify to \(\ln W \approx N \ln N - n \ln n - (N-n) \ln (N-n) + n \ln p + (N-n) \ln (1-p)\).

Differentiate \(ln W\) with respect to \(n\)

Differentiate \(\ln W\) with respect to \(n\) and set it to zero: \(\frac{d}{dn}\left[N \ln N - n \ln n - (N-n) \ln (N-n) + n \ln p + (N-n) \ln (1-p)\right] = 0\).

Solve the derivative equation

The differentiated equation is \( \ln \left(\frac{N-n}{n}\right) + \ln \left(\frac{1-p}{p}\right) = 0\). Simplify to \(\frac{N-n}{n}\cdot\frac{1-p}{p} = 1\).

Find the maximizing value of \(n\)

Solve for \(n\): \(\frac{N-n}{n} = \frac{p}{1-p}\). This gives \(N-n = n \frac{p}{1-p}\), or \(n \left(1 + \frac{p}{1-p}\right) = N\). Thus, \(n \left(\frac{1-p+p}{1-p}\right) = N\). Hence \(n^{*} = Np\).

Key concepts

Stirling's approximation

Stirling's approximation is a central mathematical tool used to approximate factorials, especially for large numbers. This approximation states that for a large integer x:
\[x! \approx \left(\frac{x}{e}\right)^x\sqrt{2\pi x}\]
For practical purposes, like in our binomial distribution problem, we often simplify to:
\[x! \approx \left(\frac{x}{e}\right)^x\]
By using this approximation, we avoid the computational complexity of factorials, making it easier to apply calculus tools like differentiation. Understanding Stirling's approximation not only simplifies our expressions but is crucial in many areas of statistics and probability.

Natural logarithm

The natural logarithm, denoted as \(\text{ln}\), is a logarithm to the base of the mathematical constant \(e\), approximately equal to 2.71828. Natural logarithms are beneficial for simplifying multiplication into addition, which is easier to differentiate.
Given our function \(\text{W}\), simplifying it involves taking its natural logarithm:
\[\ln W = \ln\left(\frac{N!}{n!(N-n)!} p^n (1-p)^{N-n}\right)\]
Utilizing properties of logarithms, this complex expression can be broken down into simpler, differentiated components. So, it's easier to handle in differentiation, which leads us to the value that maximizes our function.

Differentiation

Differentiation is a fundamental concept in calculus used to find the rate at which a function is changing at any given point. When we differentiate \ln W with respect to n:
\[\frac{d}{dn}\left[N \ln N - n \ln n - (N - n) \ln (N - n) + n \ln p + (N - n) \ln (1 - p)\right] = 0\]
This means we are looking for the point where the slope of \ln W is zero. This zero slope indicates a maximum or minimum point. In our case, we solve this derivative equation to find where our function \(\text{W}\) reaches its maximum.

Binomial distribution

The binomial distribution is a fundamental probability distribution in statistics that describes the number of successes in a fixed number of independent Bernoulli trials.
If \(W\) represents the probability mass function of a binomial distribution, where \(\text{N}\) is the number of trials, \(n\) is the number of successes, and \(p\) is the probability of success on each trial, we write:
\[W = \frac{N!}{n!(N-n)!} p^n (1-p)^{N-n}\]
To find the maximum value of \(W\), we use Stirling's approximation and natural logarithms to simplify the expression, then apply differentiation. We eventually find that the value of \(n\) which maximizes \(W\) is \(n = Np\). This insight is crucial for applications like hypothesis testing, quality control, and decision-making under uncertainty.