Problem 41 - Calculus and its applications

Question

If \(f(x)=12 x^{2}\) is a probability density function over \([-a, a],\) what is \(a ?\)

Step-by-Step Solution

Answer

The value of \(a\) is \(\frac{1}{2}\).

Step 1: Understanding the problem

For a function to be a valid probability density function (PDF), it must satisfy two main conditions: 1. It must be non-negative for all values within its support. 2. The integral of the PDF over its entire support must equal 1. Here, we need to find the value of \(a\) such that the integral of \(f(x) = 12x^2\) over \([-a, a]\) equals 1.

Step 2: Set up the integral equation

Since \(f(x) = 12x^2\) is defined over \([-a, a]\), we need to integrate it over this interval and set it equal to 1: \[ \int_{-a}^{a} 12x^2 \, dx = 1. \]

Step 3: Evaluate the integral

Evaluate the definite integral of \(12x^2\) from \(-a\) to \(a\):\[ \int_{-a}^{a} 12x^2 \, dx = \left[ 4x^3 \right]_{-a}^{a} = 4a^3 - 4(-a)^3 = 4a^3 + 4a^3 = 8a^3. \]

Step 4: Solve for \(a\)

Set the equation from the previous step equal to 1 and solve for \(a\):\[ 8a^3 = 1. \]Divide both sides by 8:\[ a^3 = \frac{1}{8}. \]Take the cube root of both sides to solve for \(a\):\[ a = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2}. \]

Key Concepts

Integral Calculus

Integral Calculus is a fundamental component of mathematical analysis that deals with the concept of integrals. It is often introduced alongside its counterpart, Differential Calculus. The primary aim of integral calculus is to determine the accumulation of quantities or the total value affected by changing variables over an interval.
The integral process typically involves two main types: indefinite integral (the antiderivative) and definite integral, which evaluates the area under a curve between specified limits. In many practical situations, you use definite integrals to compute total values, such as areas, probabilities in various distributions, or physical quantities like distance traveled.

• **Fundamental Theorem of Calculus**: This theorem links integration and differentiation, stating that integration can be reversed by differentiation.
• **Application in Probability**: Integral calculus plays an essential role in statistics, especially in the context of probability density functions (PDFs), where the integral of a function over a specific interval equals a probability.

Understanding integral calculus becomes crucial when interpreting and solving real-world problems that involve changes over a continuum.

Definite Integral

The definite integral is a key concept in calculus, crucial for computing values over a specific range. When we talk about a definite integral, we focus on the evaluation of the integral of a function over a closed interval \([a, b]\). This results in the "total" or "net" amount derived from the function over that range.
In simple terms, you use definite integrals to calculate the area under a curve bounded by the x-axis and vertical lines at \;x = a\; and \;x = b\;.

• **Evaluation Process**: To solve a definite integral, you compute the antiderivative of the function and then apply the limits with the formula \([F(b) - F(a)]\).
• **Applications**: Definite integrals are widely used in physics for finding distances, in economics for total cost or profit, and importantly, in probability for finding probabilities within a given range.

In probability density functions, the definite integral signifies the "expected" outcome over a specified interval, ensuring that the total probability sums to one. This concept was applied in the exercise to determine the span over which the function represented a valid probability.

Cube Root Calculation

The cube root calculation is a mathematical process that finds a number which, when multiplied by itself three times (cubed), equals a given number. Calculating cube roots can be understood as the reverse operation of evaluating cubes.
This concept played a role in the exercise when determining the value of \(a\) for which the definite integral meets the condition needed in a probability density function.

• **Notation**: The cube root of a number \(x\) is denoted as \(\sqrt[3]{x}\) or \(x^{1/3}\).
• **Solving Equations**: In algebraic manipulation, especially involving integrals, the cube root becomes essential when solving for unknown variables that are cubed.

In the given problem, after establishing that the integral equation is equal to one (that ensures the condition of a valid PDF), taking the cube root of \(\frac{1}{8}\) yielded \(a = \frac{1}{2}\), showing how cube roots help simplify polynomials and achieve solutions.