Problem 21 - Calculus Concepts

Question

Indicate whether the function could be a probability density function. Explain. \(f(w)=\left\\{\begin{array}{ll}1.5\left(1-w^{2}\right) & \text { when } 0 \leq w \leq 1 \\ 0 & \text { elsewhere }\end{array}\right.\)

Step-by-Step Solution

Answer

The function \( f(w) \) can be a probability density function as it is non-negative and integrates to 1 over its domain.

Step 1: Verify Non-Negativity

The function must be non-negative for all values of \( w \). For the interval \( 0 \leq w \leq 1 \), evaluate \( f(w) = 1.5(1 - w^2) \). Since \( 1 - w^2 \) is non-negative for \( w \) in \( [0, 1] \), then \( f(w) \) is non-negative on this interval. It is zero elsewhere.

Step 2: Compute the Integral Over Its Domain

To verify that \( f(w) \) is a probability density function, the integral over its entire domain must equal 1. Calculate the integral: \[\int_0^1 1.5(1-w^2)\,dw.\]

Step 3: Solve the Integral

Perform the integration: Let \( F(w) = \int 1.5(1 - w^2)\,dw = 1.5\left[w - \frac{w^3}{3}\right]. \)Evaluate it from 0 to 1:\[F(1) - F(0) = 1.5\left[1 - \frac{1}{3}\right] - 1.5\left[0 - 0\right] = 1.5 \times \frac{2}{3} = 1.\]

Step 4: Conclusion

Since \( f(w) \) is non-negative over its domain and the integral from 0 to 1 is equal to 1, \( f(w) \) satisfies the conditions for a probability density function.

Key Concepts

Non-Negativity of Probability Density Functions

A probability density function (PDF) must be non-negative. This means that its value cannot be negative for any possible input within its defined range. In simpler terms, the output of a probability density function, which might represent probabilities, must be zero or positive. For example, consider the function given in the exercise: \(f(w) = 1.5(1 - w^2)\) when \(0 \leq w \leq 1\) and 0 elsewhere. - In this interval, as \(w\) ranges from 0 to 1, the function \(1 - w^2\) remains non-negative because the square of any real number (\(w^2\)) is less than or equal to 1 in this specific range.- Thus, \(1.5(1 - w^2)\) is also non-negative. In any other range beyond 0 and 1, the function is explicitly set to zero, maintaining non-negativity. Ensuring non-negativity is crucial because probabilities cannot be negative in real-world applications, aligning mathematical consistency with practical interpretations.

Integration in Probability Theory

Integration is a fundamental tool in probability theory, especially when dealing with continuous probability distributions. To verify whether a function is a valid probability density function, one must integrate it over its entire domain of definition.For the given function \(f(w)\), the domain is from 0 to 1:\[ \int_0^1 1.5(1 - w^2)\,dw \]Integration essentially sums up all the infinitesimal probabilities over the interval. This sum (integral) should equal 1 since the total probability of all possible outcomes must be exactly 1.- The integration process involves finding the antiderivative of the function, in this case, involving \(1.5(1 - w^2)\).- After solving, if the result equals 1, it confirms the function as a valid PDF.This integration process exemplifies the ability of calculus to help express and solve practical problems like summing probabilities over a continuum.

Understanding Mathematical Functions in PDFs

Mathematical functions form the backbone of probability density functions. They help translate real-world phenomena into precise mathematical interpretations. In a PDF, the function describes how probability is distributed over a continuous random variable.- Consider \(f(w) = 1.5(1-w^2)\). Here, this quadratic function illustrates how the probability density behaves over its domain \(0 \leq w \leq 1\).- This particular function peaks at \(w = 0\) and decreases symmetrically as \(w\) approaches 1 from either side. By utilizing functions, we can model complex patterns like bell curves, exponential decays, or even uniform distributions in a manageable and analyzable form. Functions simplify the understanding of probabilistic behaviors and assist in predicting outcomes, thus bridging the gap between abstract mathematics and tangible probability applications.

Probability Theory and PDFs

Probability theory provides a systematic framework to measure how likely events are to occur. It uses tools like probability density functions to model the likelihood of different outcomes in a continuous setting.A PDF, such as \(f(w) = 1.5(1-w^2)\), helps depict the density of probabilities across a spectrum rather than discrete points. Here are some key concepts to remember about PDFs within probability theory:- **Continuous Variables:** Unlike discrete probabilities, continuous random variables can take any value within a given range.- **Area under the Curve:** The total area under the PDF curve represents the total probability, which should equal 1 over the entire range.Probability theory, using these concepts, enables the prediction and analysis of random phenomena, enhancing decision-making processes in various fields such as finance, engineering, and sciences.