Problem 7 - Applied Calculus

Question

Let \(p(x)\) be the density function for annual family income, where \(x\) is in thousands of dollars. What is the meaning of the statement \(p(70)=0.05 ?\)

Step-by-Step Solution

Answer

The density at 70,000 dollars is 0.05, indicating potential probability content but not an exact value probability.

Step 1: Understand the Concept of a Density Function

A density function, often denoted as \(p(x)\), describes how the values of a random variable, like annual family income in this case, are distributed over an interval. It's important to note that density functions are used in the context of continuous variables.

Step 2: Review Properties of a Density Function

A density function \(p(x)\) has properties: it must be non-negative for all \(x\), and the integral over the entire sample space (all possible values of \(x\)) must be equal to 1. This is because a density function represents a probability distribution over a continuous range.

Step 3: Interpret the Statement \(p(70) = 0.05\)

The statement \(p(70) = 0.05\) indicates that at \(x = 70\) (meaning \(70,000\) dollars), the density of the distribution is 0.05. This doesn't mean there's a 5% probability that a family makes exactly \(70,000\) dollars - instead, it means that the probability density at this point is 0.05.

Step 4: Relate Density to Probability

Since \(p(x)\) is a density function for a continuous variable, the actual probability of \(x\) being exactly 70,000 dollars is zero. Instead, to find the probability of income within an interval, you would integrate \(p(x)\) over that interval, not evaluate it at a single point.

Key Concepts

Probability Distribution

A probability distribution provides a comprehensive picture of where a random variable's possible outcomes might lie. In statistics, this is represented by a function, such as a density function for continuous variables like family income. This distribution helps in understanding how likely different outcomes are for a variable we are observing or measuring. By having this distribution, you can assess the probability of the random variable falling within a particular range.

• Probability distributions can be for discrete or continuous variables.
• For continuous variables, they are often represented by density functions.
• The area under the curve of the density function provides the probability for certain intervals.

Understanding probability distributions allows us to interpret and make predictions about the real-world phenomena we are studying, such as incomes in this exercise.

Continuous Variables

Continuous variables can take an infinite number of values within a given range. This is in contrast to discrete variables, which have specific or isolated values. When dealing with continuous variables, we use concepts of calculus to handle and interpret them, such as integrals and density functions.

• Common examples of continuous variables include height, weight, and in this exercise, income.
• Unlike discrete variables, probabilities for continuous variables are determined for intervals, not specific values.
• Because continuous variables can take so many values, individual outcomes have zero probability.

This understanding is crucial when considering analysis techniques for similar statistical problems.

Integral of Density Function

Integrating a density function is how we find probabilities concerning different intervals of a continuous variable, such as the income between two specific values. The integral of the density function over a range gives the probability that the variable falls within that range. This is because the integral sums up all the tiny increments of probability density, offering a total probability for the interval.

• To compute such probabilities, the density function is integrated over the desired interval of interest.
• The entire integral of the density function over all possible values must equal 1, ensuring it covers the entire probability distribution.

Thus, integration is a key tool in turning density into probability.

Probability Density

Probability density functions (PDFs) represent how dense the probability is at a specific point for continuous variables. While the value of a PDF at a specific point doesn't directly indicate probability (as probability for an exact value is zero), it shows where values cluster.

In the context of the given exercise, if the density at 70 thousand dollars is 0.05, it simply indicates the level of concentration of incomes around that value.

• Probability density is a non-negative value that provides insight into value distribution.
• High density means values are more "packed" together around the point.
• Actual probabilities require integral computation over an interval, not at a single point.

Understanding this concept aids in interpreting density functions and helps in identifying how outcomes are spread over the entire range of data.