Question

Use a program similar to the Simpson's Rule program on page 906 with \(n=6\) to approximate the indicated normal probability. The standard normal probability density function is \(f(x)=(1 / \sqrt{2 \pi}) e^{-x^{2} / 2}\). If \(x\) is chosen at random from a population with this density, then the probability that \(x\) lies in the interval \([a, b]\) is \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 1) $$

Short answer

The overall steps aim at approximating the integral of the function \(f(x)=(\frac{1}{\sqrt{2\pi}})e^{-x^2/2}\) from \([0, 1]\) using Simpson's Rule with 6 intervals. To do so, determine the interval width, calculate the function values at certain points, and then apply Simpson's Rule. The actual numerical answer depends heavily on the precise calculations in step 2.

Step-by-step solution

- Determine interval width

First, calculate the width of the intervals. The total width of the range [0,1] is simply \(1-0 = 1\). If we need to divide this into 6 equal intervals (\(n=6\)), the width of each interval (\(h\)) is given by \(\frac{1}{6}\).

- Calculate function values

Now, we need to evaluate \(f(x)\) at different points that are determined by the intervals. These values are calculated at the endpoints and at the midpoints of the intervals. Specifically, these points are: 0, 1/6, 2/6, 3/6, 4/6, 5/6, and 1. To get the function value at these points, plug in each value into the given function: \(f(x)=(\frac{1}{\sqrt{2 \pi}}) e^{-x^{2} / 2}\).

- Apply Simpson's Rule

Simpson's Rule is an approximation of the definite integral of a function. The general formula for Simpson's Rule is: \(\frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]\). Here, the terms \(f(x_0)\) and \(f(x_n)\) are the values of the function at the endpoints, while the others are values at the midpoints. To get an approximation of the definite integral, plug in the calculated function values and multiply them by the weights as indicated by the formula (odd-indexed terms are multiplied by 4, even-indexed terms by 2). The sum of these values, multiplied by \(h/3\), will give the approximated integral.

Key concepts

Standard Normal Distribution

The standard normal distribution is a specific case of the normal distribution. It is a probability distribution that is symmetrical about the mean, with a mean of 0 and a standard deviation of 1. This results in a bell-shaped curve that is used in statistics to describe how data is spread or distributed.

In any normal distribution, most of the data points are centered around the mean. For the standard normal distribution, this mean is specifically 0. Hence, it standardizes data, making it easier to calculate probabilities and compare scores from different normal distributions.

The standard normal distribution uses the probability density function (PDF) given in the exercise. This function describes the likelihood of a random variable taking on a particular value. Understanding this distribution is fundamental when calculating probabilities using tools like integration, which helps find the area under the curve between two points.

Probability Density Function

The probability density function (PDF) is a fundamental concept used to specify the probability of a continuous random variable falling within a particular range of values. For a continuous variable, such as the standard normal distribution we are dealing with, PDFs are used instead of probability mass functions, which are used for discrete variables.

The specific PDF for the standard normal distribution is \[f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}\].

This function represents how the probability of the random variable changes across different values of x—the variable in question. It informs us that large probabilities occur for values of x near the mean (0 for the standard normal distribution) and that probabilities become smaller for values further from the mean.

• The height of the PDF at any given point indicates the relative likelihood of the random variable taking on that value.
• The total area under the PDF curve is always equal to 1, as it represents the entire probability space for the random variable.

Definite Integral

The concept of a definite integral is integral (pun intended) to finding probabilities for continuous distributions. A definite integral calculates the accumulated area under a curve from one point to another. This accumulation represents the total probability that a random variable falls between those points.

In the context of this exercise, calculating \[\int_{0}^{1} f(x) \, dx\] using the definite integral gives us the probability that a randomly chosen variable, following the standard normal distribution, falls between 0 and 1.

The area under the curve is crucial because it gives a precise measure of this probability. However, calculating definite integrals analytically for complex functions like the normal distribution's PDF can be difficult.

• This is why numerical methods like Simpson's Rule are employed; they approximate the area when exact calculation is not feasible.

Numerical Approximation

Numerical approximation methods are techniques used to estimate the value of mathematical expressions that are difficult or impossible to calculate exactly. These methods are especially helpful in integrating functions for which no simple antiderivative exists, such as the normal distribution's probability density function.

In this exercise, Simpson's Rule is utilized for numerical approximation. It is a method that approximates the area under a curve (i.e., the definite integral) by fitting parabolas between sets of points on the curve.

Here's how Simpson's Rule is applied:

• The interval of integration is divided into an even number of sub-intervals (n=6 in this case).
• Function values are calculated at both endpoints and at the midpoint of each sub-interval.
• The approximation formula \[\frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]\] is used, where f(x) values at endpoints are summed once, and f(x) values at interior points are multiplied by either 4 or 2, depending on their position.

Simpson's Rule is favored for its simplicity and efficiency, requiring fewer calculations while providing acceptable accuracy for smooth functions such as the standard normal distribution.