Problem 18 - 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. $$ f(x)=k, \quad[1,4] $$

Step-by-Step Solution

Answer

The probability density function is \(f(x) = \frac{1}{3}\) over \([1, 4]\).

Step 1: Understand the Requirements of a Probability Density Function

For a function to qualify as a probability density function over an interval, the total area under the function (when integrated over the specified interval) must equal 1. In our case, the function is constant over the interval \([1,4]\).

Step 2: Integrate the Function Over the Interval

Since the function is constant, integrating it over the interval \([1,4]\) is straightforward. The integral of \(f(x) = k\) from 1 to 4 is \[\int_{1}^{4} k \, dx = k(x)]_{1}^{4}\].

Step 3: Evaluate the Definite Integral

Substitute the interval limits into the evaluated integral: \[k(x)]_{1}^{4} = k(4) - k(1) = 3k\].

Step 4: Set the Integral Equal to 1 and Solve for \(k\)

To satisfy the condition of being a probability density function, set the integral equal to 1: \[3k = 1\]. Solve this equation for \(k\) to get \(k = \frac{1}{3}\).

Step 5: Write the Probability Density Function

Now that we have determined \(k = \frac{1}{3}\), the probability density function is \[f(x) = \frac{1}{3}\] for the interval \([1,4]\).

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves calculating the area under a curve. It is essentially the process of finding an integral. In the context of probability density functions, integration helps in determining the likelihood of a random variable falling within a particular range.
When integrating a constant function like in this example, the process simplifies, as constants are easy to handle:

• Identify the interval over which to integrate. Here, it is \([1,4]\).
• Set up the integral of the function over this interval.
• Evaluate the integral using fundamental calculus rules.

For the constant function \(f(x)=k\), the integral becomes \[\int_{1}^{4} k \, dx = k(x) \biggr|_{1}^{4}\]. This operation explores how integration applies specifically to setting up and evaluating a probability density function.

Definite Integral

The definite integral is a type of integral with defined upper and lower limits, representing the total area under the curve within that specific interval. When determining if a function, such as a probability density function, integrates to a particular value, the definite integral is utilized.
In this problem, the definite integral of \(f(x)=k\) is evaluated over \[1,4\]. Substituting these limits of the interval into \[k(x)\biggr|_{1}^{4}=k(4) - k(1)=3k\], we see how we assess the function's behavior:

• The result of this definite integral must equal 1 for the function to represent a probability density function.
• Definite integrals are essential for ensuring that cumulative probabilities over an interval meet this requirement.

The definite integral brings the fixed starting point and endpoint into the equation to provide a clear value representing total probabilistic mass over the interval.

Solving Equations

A key part of this exercise involves solving an equation derived from the integration process. Once the definite integral is set up and evaluated, the next step is to solve:

• The given equation \(3k=1\) comes from equating the integral to 1 because, for probability density functions, total area = 1.
• Solving simple linear equations such as this involves isolating the variable, leading here to \(k=\frac{1}{3}\).

Equation solving here is straightforward but crucial as it silently dictates the shape and boundedness of the probability function, ensuring it behaves as expected.

Calculus Basics

Understanding calculus fundamentals is invaluable for tackling problems involving probability density functions and other mathematical concepts. Basics cover a wide span of ideas that build towards solving our specific issue:

• Grasping basic function properties helps with integrating across intervals, whether constant or variable.
• Recognizing the importance of setting up correct integrals allows solving for unknowns like \(k\) accurately.
• Comprehending linear equations is foundational to finding solutions after integration.

By building upon these foundational ideas, students can approach calculus not just as a mathematical technique but a tool for solving meaningful real-world problems, such as defining valid probability density functions.