Question

Probability The probability of finding between \(a\) and \(b\) percent iron in ore samples is modeled by \(P(a \leq x \leq b)=\int_{a}^{b} 2 x^{3} e^{x^{2}} d x, \quad 0 \leq a \leq b \leq 1\) (see figure). Find the probabilities that a sample will contain between (a) \(0 \%\) and \(25 \%\) and (b) \(50 \%\) and \(100 \%\) iron.

Short answer

The probabilities that a sample will contain between 0 and 25 percent iron and between 50 and 100 percent iron will be obtained by evaluating the integral from Step 2 and Step 3 respectively. The exact values will depend on the tools used to compute the integrals.

Step-by-step solution

Interpret the Problem Statement

The problem asks for the probability of finding between certain percentages of iron in ore samples. The probability function is given as \(P(a \leq x \leq b)=\int_{a}^{b} 2 x^{3} e^{x^{2}} dx\). The goal is to find the probabilities for two ranges of values: \([0.00,0.25]\) and \([0.50,1.00]\).

Evaluate the First Integral

For the range \([0.00,0.25]\), the corresponding probability is given as \(P(0 \leq x \leq 0.25)=\int_{0}^{0.25} 2 x^{3} e^{x^{2}} dx\). This involves integrating a function of \(x\) from \(0\) to \(0.25\). This integral can be solved using fundamental calculus techniques or with the help of computational tools.

Evaluate the Second Integral

For the range \([0.50,1.00]\), the corresponding probability is given as \(P(0.50 \leq x \leq 1)=\int_{0.50}^{1} 2 x^{3} e^{x^{2}} dx\). This involves integrating the same function of \(x\) from \(0.50\) to \(1.00\). Again, this integral can be solved using fundamental calculus techniques or with computational tools.

Key concepts

Definite Integrals

Definite integrals are fundamental in calculus as they represent the accumulation of quantities, such as area under a curve. To understand the concept, imagine plotting a curve on a graph and looking to find the area between the curve and the x-axis over a specific interval. The definite integral is a mathematical tool to do just that, standardized by the notation \[\int_{a}^{b} f(x) dx\], where \(f(x)\) is the function being integrated over the interval from \(a\) to \(b\).

In the context of probability, a definite integral can represent the probability that a random variable falls within a certain range. The result of this integration gives us the probability for continuous random variables, such as percentages of iron in an ore sample. The exercise provided uses a definite integral to model the probability of finding a certain range of iron percentages within ore samples, with the integral boundaries representing the range of interest.

Probability Density Function

A probability density function (PDF) is affiliated with continuous random variables and serves to outline the likelihood of a random variable taking on a specific value within a continuous range. Essentially, the PDF allows us to identify how probabilities are distributed across continuous outcomes.

For instance, the function \(2x^3 e^{x^2}\) in our exercise is a PDF for the content of iron in ore samples. A key point to remember is that the area under the entire curve of a PDF over its range equals 1, representing a certainty that some value within that range will occur. To find the probability that the variable falls within a sub-range \[a, b\], one must compute the integral of the PDF over \[a, b\], which is what our exercise demonstrates.

Calculus in Statistics

Calculus, particularly through the use of integrals, plays a crucial role in statistics, notably for calculating probabilities like in our exercise. Integrals help in defining the concepts such as mean, variance, and distributions of random variables. Calculus is the mathematical underpinning that allows for the precise calculations of these statistical measures.

In applied statistics, integrals are used to figure out probabilities for continuous random variables, much like summing individual probabilities in the discrete case. By integrating the probability density function over a range, as shown in the exercise, you receive probabilities that occur within that continuous range, which facilitates statistical analysis and inference.

Integration Techniques

There are several integration techniques in calculus designed to tackle a wide array of functions. Techniques such as substitution, integration by parts, and partial fractions are common. Additionally, more complex functions might require numerical methods or even software to integrate, especially when the integrand is not easily handled by standard techniques.

In the exercise solution, integrating the function \(2x^3 e^{x^2}\) could require substitution where one would let \(u=x^2\), making \(du = 2x dx\). However, since \(2x^3\) is not a direct derivative of \(x^2\), this integral may need more advanced methods or computational tools for an accurate evaluation. Understanding various integration techniques is essential for solving real-world problems, as they can supply exact or approximate solutions to complex integrals found in scientific, engineering, and statistical applications.