Problem 4 - Calculus for Biology and Medicine

Question

Determine \(c\) such that \(f(x)=\left\\{\begin{array}{ll}\frac{c}{x^{2}} & \text { for } x>1 \\ 0 & \text { for } x \leq 1\end{array}\right.\) is a density function.

Step-by-Step Solution

Answer

The value of \( c \) is 1.

Step 1: Understand the Requirements for a Density Function

A probability density function (PDF) must satisfy two main conditions: 1. The function must be non-negative for all possible values of the random variable. 2. The integral of the function over its entire range must equal 1.

Step 2: Determine the Range of Integration

Since the given function is defined as \( f(x) = \frac{c}{x^2} \) for \( x > 1 \), the function is non-zero starting from \( x = 1 \). Therefore, the range of integration for this function will be from 1 to infinity.

Step 3: Set Up the Integral

The integral of the function over its defined range must equal 1. Set up the integral equation:\[\int_{1}^{ ext{∞}} \frac{c}{x^2} \, dx = 1\]

Step 4: Solve the Integral

Calculate the integral \( \int \frac{c}{x^2} \, dx \). Use the integral formula \( \int x^{-n} \, dx = \frac{x^{-n+1}}{-n+1} + C \) for \( n eq 1 \):\[\int \frac{c}{x^2} \, dx = c \left( -\frac{1}{x} \right) + C\]Evaluate from 1 to infinity.

Step 5: Evaluate the Integral from Limits

Evaluate \( \left. -\frac{c}{x} \right|_1^{\infty} = 0 - (-c) = c \). Therefore, we have the equation:\[ c = 1 \]

Step 6: Conclusion

Since \( c = 1 \) satisfies the condition that the integral from 1 to infinity of \( f(x) \) is equal to 1, the correct value of \( c \) that makes \( f(x) \) a probability density function is 1.

Key Concepts

Understanding Probability Distribution

A probability distribution is a fundamental concept in statistics that describes how probabilities are assigned to different outcomes of a random variable. It provides a comprehensive way to understand the likelihood of various possible outcomes. In the case of continuous variables, such as the one given in the exercise, we use a probability density function (PDF). A PDF tells us the relative likelihood of a random variable falling within a certain range of values. This is crucial, as it helps us model real-world data and perform statistical analysis.

There are two essential criteria a PDF must meet:

• The function must be non-negative over its entire range. This ensures that probability values, which are inherently non-negative, are accurately represented.
• The total area under the curve of the PDF must equal 1. This reflects the certainty that the random variable assumes some value within its range.

In our exercise, the goal is to determine the correct constant value that makes the given function a valid PDF. This requires ensuring that when integrated over its range, the function satisfies the total probability condition.

Exploring Integral Calculus

Integral calculus is a branch of mathematics that deals with the concept of integration, which is essentially the opposite of differentiation. Integration helps us find areas under curves, which is critical when working with probability distributions. In this context, integration allows us to verify the total probability of a distribution function by summing the infinitesimally small probabilities over the continuous range.

The problem in the exercise involves an integral over a probability density function to ensure that the total area under the curve equals 1. By defining the integration limits from 1 to infinity, we focus on the domain where the function is non-zero. This step emphasizes the application of integral calculus to verify that our probability distribution meets its necessary criteria.

Application of Function Integration

Function integration is a process of finding the integral of a function, which in mathematics, is related to finding the 6sum" of infinitely tiny quantities. In the context of this exercise, integrating the given function over its defined range allows us to calculate the cumulative probability.

• To solve the integral \( \int_{1}^{\infty} \frac{c}{x^2} \, dx = 1 \), recognize the pattern in the function \( \frac{c}{x^2} \).
• Use the power rule for integration, which states that \( \int x^{-n} \, dx = \frac{x^{-n+1}}{-n+1} + C \), where \( n eq 1 \).
• Thus, the integral becomes \( -\frac{c}{x} \), evaluated from 1 to infinity, simplifying to \( c = 1 \).

Function integration thus plays a key role in validating that our function acts appropriately as a probability density function by ensuring the total probability equals 1. This highlights the importance of integration in creating and verifying models in probability and statistics.