Problem 140 - An Introduction to Mathematical Statistics and Its Applications

Question

Let \(X\) be the time in days between a car accident and reporting a claim to the insurance company. Let \(Y\) be the time in days between the report and payment of the claim. Suppose that \(f_{X, Y}(x, y)=c, 0 \leq x \leq 7,0 \leq y \leq 7\), and zero otherwise. (a) Find \(c\). (b) Find \(P(0 \leq X \leq 2,0 \leq Y \leq 4)\).

Step-by-Step Solution

Answer

\(c = \frac{1}{49}\), \(P(0 \leq X \leq 2, 0 \leq Y \leq 4) = \frac{8}{49}\)

Step 1: Normalize the joint Probability Density Function

The joint pdf, \(f_{X, Y}(x, y)\) is equal to c, on a square of \(0 \leq x, y \leq 7\), so to normalize the function we need to solve for \(c\). The total probability must equal 1 over all values of X and Y. So we can set up and solve the following equation: \(c \cdot \int_{0}^{7} \int_{0}^{7} dx \, dy = 1\). Simplified this becomes: \(c \cdot (7)^2 = 1 \).

Step 2: Solve for \(c\)

Solve the equation \(c \cdot (7)^2 = 1\) for \(c\). After solving, we will see that \(c = \frac{1}{49}\).

Step 3: Calculate the joint probability

The problem now asks for \(P(0 \leq X \leq 2, 0 \leq Y \leq 4)\). This probability can be calculated as: \[\int_{0}^{2} \int_{0}^{4} f_{X, Y}(x, y) \, dx \, dy \]. After substituting the value of \(c\) and the limits, the calculation simplifies to: \[ \frac{1}{49} \cdot 2 \cdot 4 \]

Step 4: Solve for the joint probability

Finally, solve \(\frac{1}{49} \cdot 2 \cdot 4\) and this yields \(P(0 \leq X \leq 2, 0 \leq Y \leq 4) = \frac{8}{49}\)

Key Concepts

Probability Density Function

A probability density function (PDF) is fundamental in understanding the likelihood of different outcomes within a continuous random variable. If we were visualizing the concept, we could imagine a smooth curve or surface that describes the relative chances of different outcomes. The higher the curve or surface at a particular point, the greater the likelihood of that outcome.

In the case of joint PDFs, like in the exercise with variables X (days to report an accident) and Y (days to pay a claim), the function represents the probability that the random variables fall within a specific range. The constant 'c' that appears in the textbook exercise is part of the formula for the joint PDF, which must integrate to 1 over the range of possible values, indicating that the total probability across all outcomes is certain.

Normalization of Probability

Normalization is a fundamental technique in probability and statistics. It ensures that the probabilities over all possible outcomes add up to one. When it comes to PDFs, normalization means finding the constant that makes the integral of the PDF over its entire space equal to one.

Think of it as adjusting the 'volume' under the surface defined by the joint PDF so that it fills a 'probability container' perfectly, without any spillover or shortfall. In our exercise, finding the value of 'c' was necessary to make sure that the total probability over the square region from 0 to 7 days for both X and Y variables sums to one.

Double Integration in Probability

When dealing with multiple continuous random variables, as in the exercise with X and Y, double integration becomes a vital tool. Double integration allows us to calculate the probability of events occurring within a certain region for two random variables. We sum over the infinitesimally small areas within the bounds, multiplying by the joint PDF to obtain the total probability.

In simpler terms, double integration helps us add up all the tiny probabilities across the surface described by the joint PDF. This is how we found the probability for the event where 0 ≤ X ≤ 2 and 0 ≤ Y ≤ 4. By integrating the PDF over this region, we determined the chance of both X and Y falling within these timeframes.

Mathematical Statistics Concepts

Mathematical statistics involves a suite of concepts and procedures to interpret and analyze probabilistic events quantitatively. Concepts like expectation, variance, hypothesis testing, and regression analysis fall under this broad umbrella.

In our textbook problem, we've touched upon several of these concepts: the nature of random variables, their distributions (captured by the PDF), and finding specific probabilities. It’s these kinds of tools that allow statisticians to make informed decisions based on data, ranging from insurance risks to medical trials and beyond. Understanding the joint probability between X and Y in terms of PDFs is just one of the multifaceted approaches in this field of study.