Problem 7 - Calculus

Question

If you choose \(x\) completely at random between 0 and \(\pi\) what is the density \(p(x)\) and the cumulative density \(F(x)\) ?

Step-by-Step Solution

Answer

\( p(x) = \frac{1}{\pi} \), \( F(x) = \frac{x}{\pi} \) for \( x \) in \( [0, \pi] \).

Step 1: Understanding Uniform Distribution

If a variable is chosen randomly within a certain interval, it is uniformly distributed over that interval. Here, we are choosing \( x \) completely random between \( 0 \) and \( \pi \), indicating a uniform distribution over \( [0, \pi] \).

Step 2: Calculating the Probability Density Function (PDF)

For a uniform distribution from \( a \) to \( b \), the probability density function \( p(x) \) is given by \( \frac{1}{b-a} \) for \( x \) in \( [a,b] \). Here, \( a = 0 \) and \( b = \pi \), so we have \( p(x) = \frac{1}{\pi-0} = \frac{1}{\pi} \).

Step 3: Deriving the Cumulative Distribution Function (CDF)

The cumulative distribution function \( F(x) \) for a uniform distribution is calculated by integrating the PDF from the lower bound up to \( x \). Therefore, \( F(x) = \int_0^x \frac{1}{\pi} \, dt = \frac{x}{\pi} \), where \( t \) is a dummy variable of integration.

Step 4: Final Results Compilation

The density function is \( p(x) = \frac{1}{\pi} \) and the cumulative density function is \( F(x) = \frac{x}{\pi} \) for \( x \) in \([0, \pi]\).

Key Concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in statistics, particularly when dealing with continuous random variables like the uniform distribution in our example. The PDF describes the likelihood of a random variable to take on a particular value. For a uniform distribution between two points, the PDF is constant because each outcome within the interval is equally likely.

In our exercise, since the variable is uniform between 0 and \( \pi \), the PDF is defined as:

• \( p(x) = \frac{1}{b-a} \) for \( x \) in \([a,b]\)
• Here, \( a = 0 \) and \( b = \pi \).
• So, the PDF simplifies to \( p(x) = \frac{1}{\pi} \).

This constant value of the PDF across the interval implies that all values from 0 to \( \pi \) are equally probable. Understanding the PDF helps in identifying how probability is spread across the possible values of a random variable.

Cumulative Distribution Function

The cumulative distribution function (CDF) is an essential tool for determining the probability that a continuous random variable takes on a value less than or equal to a specific point. It accumulates the probabilities, starting from the lower limit of the interval to the value of interest.

For a uniform distribution, the CDF is a linear function of \( x \). It is computed by integrating the PDF from the lower boundary up to \( x \). The process involves:

• Integrating the constant PDF, \( \frac{1}{\pi} \), from 0 to \( x \).
• The CDF formula becomes \( F(x) = \int_0^x \frac{1}{\pi} \, dt = \frac{x}{\pi} \).

The slope of the CDF is 1 divided by the interval length (\( \pi \) in this case), representing a steadily increasing probability as the variable values increase. The CDF effectively tells us the probability that the random variable is less than or equal to \( x \), thereby offering insights into the distribution’s cumulative property.

Random Variable

A random variable is a numerical outcome of a random phenomenon. In the context of the exercise, the random variable is the value \( x \) that is selected randomly between 0 and \( \pi \). The uniform distribution implies that \( x \) can take any value in this interval with equal probability.

Some important aspects of random variables include:

• They quantify uncertainty by assigning numbers to outcomes of random processes.
• Random variables can be discrete (specific, separated values) or continuous (any value within a range).
• In our example, \( x \) is a continuous random variable, since it can assume any real number from 0 to \( \pi \).

Understanding random variables is crucial as it forms the foundation for further statistical analysis and probability calculations. They help in assessing real-world situations by modeling random processes mathematically.

Integration in Calculus

Integration in calculus plays a vital role in finding cumulative properties from given rates of change, such as deriving the CDF from the PDF. It involves calculating the integral of a function over a specific interval, which essentially accumulates values of the function along that interval.

In our exercise:

• We begin with the PDF, \( \frac{1}{\pi} \), a constant function over the interval \([0, \pi]\).
• The integral of the PDF from 0 to \( x \) provides the CDF, \( F(x) = \frac{x}{\pi} \).

Integration is often seen as the reverse process of differentiation and is a powerful tool to solve problems in areas ranging from probability, like finding a CDF, to physics and engineering, where it calculates areas, volumes, and other properties related to change. Mastery of integration techniques is crucial for understanding and solving a wide range of both elementary and advanced mathematical problems.