Problem 30 - An Introduction to Mathematical Statistics and Its Applications

Question

Show that the ratio of two successive hypergeometric probability terms satisfies the following equation, $$ \frac{\left(\begin{array}{c} r \\ k+1 \end{array}\right)\left(\begin{array}{c} w \\ n-k-1 \end{array}\right)}{\left(\begin{array}{l} N \\ n \end{array}\right)} \div \frac{\left(\begin{array}{l} r \\ k \end{array}\right)\left(\begin{array}{c} w \\ n-k \end{array}\right)}{\left(\begin{array}{c} N \\ n \end{array}\right)}=\frac{n-k}{k+1} \cdot \frac{r-k}{w-n+k+1} $$ for any \(k\) where both numerators are defined.

Step-by-Step Solution

Answer

Simplify the ratio by dividing the fractions and use factorials to simplify more, primarily making use of the property \( (k+1)! = (k+1) . k!\). The simplification will lead you to the required equation. The key step in this problem involves simplifying the original equation using properties of combinations.

Step 1: Rewrite the ratio using properties of combinations

Rewrite the complex ratio equation involving binomial coefficients into a simpler form. This can be done by dividing the fraction on the top with the fraction at the bottom to start with. Remember that \(\binom{n}{k}= \frac{n!}{k!(n-k)!}\) where \(n!\) is the factorial of \(n\), \(k!\) is the factorial of \(k\) and \((n−k)!\) is the factorial of \((n−k)\).

Step 2: Simplify the equation

Simplify the equation obtained after rewriting. You can get rid of \(\binom{N}{n}\) which is common in both numerator and denominator. Then, remembering the property of factorial, \( (k+1)! = (k+1) \cdot k!\), simplify the factorials.

Step 3: Calculating the final equation

After simplifying, you would get four fractions. The numerators on the right hand side are multiplied (and denominators as well) since the original terms are in a product. Finally, simplify these fractions to get the required equation.

Key Concepts

Binomial Coefficient

A binomial coefficient is a mathematical term that shows the number of ways to choose a subset of items from a larger set, without considering the order of selection. It's like asking the question, "How many ways can I choose 3 apples from a basket of 5?" When you read or hear about binomial coefficients, you're likely to see notation like \(\binom{n}{k}\). This means choosing \(k\) items from \(n\) available items.

• The formula for calculating a binomial coefficient is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• The exclamation point (!), known as a factorial, means to multiply a series of descending natural numbers.

Binomial coefficients are fundamental in probability and statistics because they calculate probabilities in binomial distributions, which involve two possible outcomes (success or failure). This formula is vital for solving complex problems, such as the one we're improving here, by breaking down combinations into more manageable pieces.

Factorial

Factorials are a mathematical way to describe the product of an integer and all the integers below it. When you see the symbol \(n!\), it means you should multiply \(n\) by every integer less than \(n\) down to 1. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow very quickly as the numbers increase.

• Factorials are used in a variety of mathematical contexts, including permutations, combinations, and algebraic equations.
• They help in simplifying expressions that involve binomial coefficients, as seen in the binomial coefficient formula.
• Understanding the properties of factorials, like \((k+1)! = (k+1) \cdot k!\), makes it easier to break down and solve complex problems.

In probability and statistics, factorials help calculate the exact number of ways an event can occur, which is crucial for finding probabilities.

Probability Distribution

Probability distribution is a key concept in statistics that shows how probabilities are distributed over different possible outcomes. One common type is the hypergeometric distribution, which is the focus of our original exercise.

• Unlike a simple distribution with independent trials (like flipping a coin), a hypergeometric distribution deals with dependent events, where each draw affects the outcome of the next.
• The formula for hypergeometric distribution often involves binomial coefficients because you're determining the number of ways of picking items, which changes as items are chosen.
• Understanding probability distributions can help in predicting the likelihood of different outcomes and enabling decision-making based on statistical models.

This distribution is particularly useful when you're interested in probabilities under constraints. For example, you might want to know the probability of drawing a certain number of colored balls from a bag without replacement. This is intertwined with understanding and calculating the ratios in hypergeometric terms, as shown in the exercise solution.