Problem 16 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

In the 18th century Georges-Louis Leclerc, Comte de Buffon (17071788 ) found an amusing way to approximate the number \(\pi\) using probability theory and statistics. Buffon had the following idea: take a needle and a large sheet of paper, and draw horizontal lines that are a needle-length apart. Throw the needle a number of times (say \(n\) times) on the sheet, and count how often it hits one of the horizontal lines. Say this number is \(s_{n}\), then \(s_{n}\) is the realization of a \(\operatorname{Bin}(n, p)\) distributed random variable \(S_{n}\). Here \(p\) is the probability that the needle hits one of the horizontal lines. In Exercise \(9.20\) you found that \(p=2 / \pi\). Show that $$ T=\frac{2 n}{S_{n}} $$ is the maximum likelihood estimator for \(\pi\).

Step-by-Step Solution

Answer

The estimator \( T = \frac{2n}{S_n} \) is the maximum likelihood estimator for \( \pi \).

Step 1: Understand the problem

Analyze the problem statement. You have a needle throwing experiment, in which you throw a needle on a sheet with parallel lines spaced a needle-length apart. Here, the number of times the needle hits a line is a binomial random variable with parameter \( p = \frac{2}{\pi} \). The task is to show that \( T = \frac{2n}{S_n} \), where \( n \) is the number of throws and \( S_n \) the number of hits, is the maximum likelihood estimator for \( \pi \).

Step 2: Write the Likelihood Function

For a binomial distribution with parameters \( n \) (the number of trials) and \( p \) (the probability of success), the likelihood function is given by:\[ L(p; S_n) = \binom{n}{S_n} p^{S_n} (1-p)^{n-S_n} \]Here, \( S_n \) is the observed number of times the needle hits a line.

Step 3: Replace \( p \) with \( \frac{2}{\pi} \)

Since we know from Exercise 9.20 that \( p = \frac{2}{\pi} \), substitute this into the likelihood function:\[ L\left(\frac{2}{\pi}; S_n\right) = \binom{n}{S_n} \left(\frac{2}{\pi}\right)^{S_n} \left(1-\frac{2}{\pi}\right)^{n-S_n} \]

Step 4: Find Maximum Likelihood Estimator

To find the maximum likelihood estimator, you need to find the value of \( \pi \) that maximizes the likelihood function. It often involves taking the derivative of the log-likelihood and setting it to zero. Start by simplifying the log-likelihood function:\[ \log L\left(\frac{2}{\pi}; S_n\right) = \log \binom{n}{S_n} + S_n \log\left(\frac{2}{\pi}\right) + (n-S_n)\log\left(1 - \frac{2}{\pi}\right) \]

Step 5: Differentiate and Solve for \( \pi \)

Differentiate the log-likelihood with respect to \( \pi \) and set the derivative to zero to find the estimator for \( \pi \). Simplification will lead you to:\[ \frac{S_n}{2/\pi} - \frac{n-S_n}{1-2/\pi} = 0 \]Solving for \( \pi \) gives:\[ \pi = \frac{2n}{S_n} \]

Step 6: Conclude Maximum Likelihood Estimator

The simplification shows that the expression \( T = \frac{2n}{S_n} \) indeed maximizes the likelihood function, hence it is the maximum likelihood estimator for \( \pi \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It's like trying to predict the future based on past data. In probability theory, we often talk about experiments, outcomes, and events. Each outcome of an experiment is associated with a probability, which is a number between 0 and 1. A probability of 0 means the event will not occur, while a probability of 1 indicates the event is certain.

Probability theory is foundational for various applications, including risk assessment, decision-making, and statistical inference. It helps us make informed guesses in uncertain situations. In the case of Buffon's needle problem, probability theory helps us understand how likely it is for the needle to hit a line on the paper based on numerous trials.

Binomial Distribution

The binomial distribution is a major concept in probability theory. It models the number of successes in a fixed number of trials. Each trial is independent and has two possible outcomes: success or failure. The probability of success, denoted as \( p \), stays the same in each trial.

In Buffon's needle, the experiment is set up to have clearly defined successes (needle hitting a line) and failures (needle missing a line). It can follow a binomial distribution with \( n \) trials and a success probability \( p = \frac{2}{\pi} \). If \( S_n \) is the number of hits observed, it follows a \( Binomial(n, p) \) distribution. This captures how likely each possible number of needle hits is, given the probability of a hit in one throw is \( \frac{2}{\pi} \), derived from the geometric setup of the lines and needle.

Buffon's Needle

Buffon's needle is a classical probability problem that approximates the value of \( \pi \). Imagine dropping a needle onto a sheet with parallel lines spaced evenly apart, each as long as the needle itself. The question becomes: "How often will the needle land crossing a line?"

Buffon's needle is fascinating because it connects a seemingly random experiment with the mathematical constant \( \pi \). The probability that the needle will cross a line, when lines are spaced by the length of the needle, is \( 2/\pi \). By performing the experiment many times and recording the number of times the needle crosses a line compared to how many times it was thrown, one can estimate \( \pi \) via statistical means. The historical context of this problem, originating in the 18th century, underscores its enduring significance in illustrating the power of probability and geometry.

Statistical Inference

Statistical inference is the process of drawing conclusions about a population's characteristics based on a sample. It allows us to make educated guesses about what we haven't observed directly, using the data we do have.

Maximum Likelihood Estimation (MLE) is a key method in statistical inference. It's about finding the parameter values that make the observed data most probable. In Buffon's needle experiment, applying MLE involves determining the value for \( \pi \) that maximizes the likelihood function based on our observations of needle hits. This leads to the formula \( \frac{2n}{S_n} \), which serves as the maximum likelihood estimator for \( \pi \). By considering all possible outcomes and their likelihoods, statistical inference helps us best use observed data to understand underlying phenomena.