Question

Use Simpson's rule with four subdivisions to approximate the area under the probability density function \(y=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}\) from \(x=0\) to \(x=0.4\)

Short answer

The approximate area using Simpson's rule with four subdivisions is around 0.1576.

Step-by-step solution

Define the Function and Interval

The function given is the probability density function \( y = \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} \). We are asked to approximate the area under this curve from \( x = 0 \) to \( x = 0.4 \).

Determine the Number of Subintervals

Simpson's rule requires us to divide the interval into an even number of subintervals. Here, it is specified to use four subdivisions. Let \( n = 4 \). The width of each subinterval \( h \) is calculated by \( h = \frac{b-a}{n} = \frac{0.4 - 0}{4} = 0.1 \).

Calculate the Function Values at the Subdivision Points

Using the subinterval width \( h = 0.1 \), we determine the \( x \) values at which we will evaluate the function: \( x_0 = 0 \), \( x_1 = 0.1 \), \( x_2 = 0.2 \), \( x_3 = 0.3 \), and \( x_4 = 0.4 \). We calculate the function values: \( f(x_0) = \frac{1}{\sqrt{2 \pi}} \), \( f(x_1) = \frac{1}{\sqrt{2 \pi}} e^{-0.1^2/2} \), \( f(x_2) = \frac{1}{\sqrt{2 \pi}} e^{-0.2^2/2} \), \( f(x_3) = \frac{1}{\sqrt{2 \pi}} e^{-0.3^2/2} \), \( f(x_4) = \frac{1}{\sqrt{2 \pi}} e^{-0.4^2/2} \).

Apply Simpson's Rule Formula

Simpson's rule formula is given by \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right) \]. Substituting our values and maintaining consistency with the formula, we calculate the approximation.

Compute the Approximate Integral

Substitute the function values from Step 3 into the Simpson's rule formula: \( \int_{0}^{0.4} \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} \, dx \approx \frac{0.1}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)) \). This calculation yields an approximate value for the integral, which is the area under the curve from \( x = 0 \) to \( x = 0.4 \).

Calculate the Final Result

Perform the arithmetic using the function values: \[ \frac{0.1}{3} \left( f(0) + 4f(0.1) + 2f(0.2) + 4f(0.3) + f(0.4) \right) = \frac{0.1}{3} \left( \frac{1}{\sqrt{2\pi}} + 4 \frac{1}{\sqrt{2\pi}} e^{-0.1^2/2} + 2 \frac{1}{\sqrt{2\pi}} e^{-0.2^2/2} + 4 \frac{1}{\sqrt{2\pi}} e^{-0.3^2/2} + \frac{1}{\sqrt{2\pi}} e^{-0.4^2/2} \right) \]. Calculate this to find the approximate integral.

Key concepts

Probability Density Function

A probability density function (PDF) is a fundamental concept in statistics and probability theory. It is used to define the probability distribution for continuous random variables. The graphical representation of a PDF is a continuous curve. It shows where a random variable is most likely to occur based on probabilities for different outcomes.
In the original exercise, the PDF is given by the function \( y = \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} \), which is a part of the Gaussian distribution, commonly known as the normal distribution. This specific PDF is critical for describing how data values are distributed in a bell-shaped curve where the mean is 0 and variance is 1. The task is to find the area under this curve from \( x = 0 \) to \( x = 0.4 \), representing the probability that the random variable takes a value within this range.

Numerical Integration

Numerical integration is a mathematical tool used when finding exact integrals of functions analytically is difficult or impossible. Instead of calculating an integral through traditional calculus methods, numerical integration methods approximate the value of the integral using discrete data points.
Simpson's Rule, which is applied in the original exercise, is one such method. It leverages parabolic arcs instead of straight-line segments to estimate the area under a curve. By using this approach, it often provides more accurate results for many functions compared to other methods like the Trapezoidal Rule. In the case of the given PDF, numerical integration serves to approximate the probability for a continuous distribution over a specified range.

Approximate Integral Calculation

Approximate integral calculation is a process wherein the integral of a complex function is estimated using methods like Simpson's Rule. Since certain functions might not possess simple antiderivatives, we rely on approximation techniques to estimate the areas beneath these functions.
The exercise illustrates this by demanding the area under the probability density function \( y = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \) between two points. By breaking the interval \([0, 0.4]\) into smaller, manageable sections, the function's values at specific \( x \)-coordinates are used. These values and their coefficients (1, 4, and 2) as dictated by Simpson’s Rule are multiplied and summed to reach an estimated integral value.

Subdivisions in Numerical Methods

In numerical methods, subdivisions play a vital role by partitioning a given interval into smaller, equal segments to perform calculations more efficiently and accurately.
For Simpson's Rule, as applied in the task, the interval \([0, 0.4]\) is divided into four equal subintervals. Each subinterval is 0.1 units wide, which is determined by the formula \( h = \frac{b-a}{n} \), where \( n \) is the number of subdivisions. By using several smaller segments, errors are minimized due to the more precise fitting of parabolic arcs through smaller sections of the function's curve.
This detailed division approach enhances the estimation accuracy and helps provide a closer approximation to the actual integral value.