Problem 135 - An Introduction to Mathematical Statistics and Its Applications

Question

An urn contains four red chips, three white chips, and two blue chips. A random sample of size 3 is drawn without replacement. Let \(X\) denote the number of white chips in the sample and \(Y\) the number of blue chips. Write a formula for the joint pdf of \(X\) and \(Y\).

Step-by-Step Solution

Answer

The joint pdf of \(X\) and \(Y\) is given by: \[ f(x, y) = \frac{{3 \choose x} {2 \choose y} {4 \choose {3-x-y}}}{{9 \choose 3}} \]for \((x, y)\) in the set \{\(0,0\), \(0,1\), \(0,2\), \(1,0\), \(1,1\), \(2,0\), \(3,0\)\}.

Step 1: Denote the Total Number of Chips and Sample Size

In this case, the total number of chips \(N\) is 9 (4 red, 3 white, 2 blue), and the sample size \(n\) is 3. Thus, the total number of ways to draw a sample of 3 out of 9 is given by the combination \({9 \choose 3}\).

Step 2: Define the Range of X and Y

Let the random variable \(X\) be the number of white chips in the sample and \(Y\) the number of blue chips. Since the sample size is 3, the possible values for \(X\) and \(Y\) are 0, 1, 2, 3. However, because \((X+Y)\) cannot exceed the sample size of 3, the possible pairs of \(X, Y\) are \((0,0), (0,1), (0,2), (1,0), (1,1), (2,0), (3,0)\).

Step 3: Write the Joint Probability Formula for Each Pair (X, Y)

The probability that X is equal to \(x\) and Y is equal to \(y\) is given by the joint pdf formula: \[ f(x, y) = \frac{{3 \choose x} {2 \choose y} {4 \choose {3-x-y}}}{{9 \choose 3}} \] This formula represents the number of ways to choose \(x\) whites out of 3, \(y\) blues out of 2 and \((3-x-y)\) reds out of 4, divided by the total number of ways to choose 3 out of 9.

Key Concepts

Probability Distribution

Probability distributions describe how probabilities are assigned to different outcomes of a random variable. In our exercise, we deal with a joint probability distribution, where two random variables, \(X\) and \(Y\), are involved. Here, \(X\) denotes the number of white chips and \(Y\) the number of blue chips chosen from the urn.

A joint probability distribution is a function \(f(x, y)\) which defines the likelihood that \(X = x\) and \(Y = y\) simultaneously. To calculate this likelihood, we consider all combinations of outcomes and determine their probabilities within the given constraints—including the total sample size and the number of chips available in the urn. By understanding the distribution, you can predict the behavior of \(X\) and \(Y\) and make informed assumptions about future draws or experiments.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting and arranging objects. It is crucial in probability for determining the number of ways an event can occur. In the given exercise, combinatorics helps us calculate possible combinations for drawing chips from the urn.

To find the number of ways to draw the chips, we use combination formulas, represented as \({n \choose k}\), which denote the number of ways to choose \(k\) items from \(n\) items without regard to order. In our solution, the total ways to draw any 3 chips from 9 is \({9 \choose 3}\). Similarly, combinations are calculated for the white, blue, and red chips separately to form the joint probability distribution formula.

Discrete Random Variables

Discrete random variables take on a countable number of distinct values. In this problem, both \(X\) and \(Y\) are discrete random variables since \(X\) represents the number of white chips and \(Y\) represents the number of blue chips drawn in the sample.

For a discrete joint distribution, it's important to identify all possible combinations of values these random variables can take. In our scenario, we determined all possible pairs \((X, Y)\) considering the physical constraints of the urn—i.e., the total sample size and the possible chips available. This understanding helps in setting up the ranges and possible outcomes for \(X\) and \(Y\), enabling more precise distribution models.