Problem 22 - Calculus and its applications

Question

Find \(k\) such that each function is a probability density function over the given interval. Then write the probability density function. $$ f(x)=\frac{k}{x}, \quad[1,3] $$

Step-by-Step Solution

Answer

The function is a PDF with \( k = \frac{1}{\ln 3} \), and the PDF is \( f(x) = \frac{1}{\ln 3} \cdot \frac{1}{x} \) on \([1,3]\).

Step 1: Understand Probability Density Functions

A probability density function (PDF) must integrate to 1 over its entire interval. For the function whose PDF we are finding, this means integrating from 1 to 3.

Step 2: Set Up the Equation

For the function \[f(x) = \frac{k}{x}, \]we need to set up the equation for the integral over the interval \([1,3]\):\[\int_{1}^{3} \frac{k}{x} \, dx = 1.\]

Step 3: Solve the Integral

Integrate \(\frac{k}{x}\) with respect to \(x\) from 1 to 3:\[\int \frac{k}{x} \, dx = k \cdot \ln|x| + C.\]Evaluating from 1 to 3:\[\left[k \ln|x| \right]_{1}^{3} = k(\ln 3 - \ln 1).\] Since \(\ln 1 = 0\), this simplifies to:\[k \ln 3.\]

Step 4: Solve for k

Set up the equation we derived to equal 1:\[k \ln 3 = 1.\]Solve for \(k\) by dividing both sides by \(\ln 3\):\[k = \frac{1}{\ln 3}.\]

Step 5: Write the Probability Density Function

Now that we have \(k\), we can write the probability density function:\[f(x) = \frac{1}{\ln 3} \cdot \frac{1}{x}, \quad [1, 3].\]

Key Concepts

Integration

Integrating a function is like finding the area underneath its curve over a given interval. This concept is key when working with probability density functions (PDFs) because a PDF must integrate to 1 over its defined interval. In this exercise, the goal is to integrate the function \(f(x)=\frac{k}{x}\) over the interval \([1,3]\).Here's a simple guide for visualizing and calculating the integral:

• Imagine drawing a graph of \(\frac{k}{x}\) from \([1,3]\).
• You're looking to find the total area between the curve, the x-axis, and the vertical lines \(x=1\) and \(x=3\).
• This area represents the total probability, which must equal 1 for the function to be a legitimate PDF.

To find the integral, use the antiderivative of \(\frac{k}{x}\), which is \(k \cdot \ln|x|\). By applying the limits of integration from 1 to 3, we can finally solve for any constants involved, such as \(k\).

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a special mathematical function that appears frequently in calculus and probability. It is the inverse function to exponentiation with the base \(e\), roughly equal to 2.71828.In the context of our exercise, natural logarithms appear when integrating fractions with linear denominators. Specifically, when you integrate \(\frac{k}{x}\), the result includes \(k \cdot \ln|x|\).Here are some fundamental properties of natural logarithms:

• The natural logarithm of 1, \(\ln(1)=0\), is critical to simplifying integrated functions.
• It has useful properties, such as \(\ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right)\), which helps simplify expressions when evaluating integrals across bounds.
• The derivative of \(\ln(x)\) is \(\frac{1}{x}\), making it the perfect partner for integration with functions like \(\frac{1}{x}\).

The understanding and usage of natural logarithms are essential when dealing with solutions that involve infinitesimally small changes, such as computing continuous probabilities.

Solving Equations

Solving equations is a critical step in mathematical problem solving, especially when tasked with finding constants that ensure a function meets certain criteria, like a probability density function integrating to 1.In our problem, we set up the equation from the integrated result: \(k \ln 3 = 1\). Solving for \(k\) involves simple algebraic steps, but they can greatly affect the final outcome:

• First, acknowledge the equation, \(k \ln 3 = 1\). We need \(k\) because it ensures the total area under the curve \(\frac{k}{x}\) from 1 to 3 is exactly equal to 1.
• To isolate \(k\), divide both sides by \(\ln 3\), resulting in \(k = \frac{1}{\ln 3}\).
• This value of \(k\) must be substituted back into the original function to accurately detail the PDF as \(f(x) = \frac{1}{\ln 3} \cdot \frac{1}{x}\).

Through strategic rearrangement and simplification, these steps culminate in a properly integrated PDF, ensuring it abides by all rules for continuous probability functions.