Problem 30 - College Algebra and Calculus: An Applied Approach

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 \(P(a \leq x \leq b)=\int_{a}^{b} f(x) d x\). $$ P(0 \leq x \leq 2) $$

Step-by-Step Solution

Answer

The solution requires applying Simpson's rule to the integral of the pdf from 0 to 2. The specific numerical answer will depend on the actual computations.

Step 1: Understanding the Simpson's Rule

Simpson’s 1/3 rule estimates the area under a curve by summing the areas of the trapezoids formed by the curve and the x-axis. Simpson’s rule applies this concept to parabolic sections of the curve, rather than linear. In mathematical form, the rule is given by: \(\int_{a}^{b} f(x) dx ≈ (b - a) / 6 [f(a) + 4f((a + b) / 2) + f(b)]\).

Step 2: Understanding the Problem

Here, the task is to use the Simpson’s rule to estimate the probability that a randomly chosen \(x\) from the given pdf will lie in the interval [0, 2]. Therefore, the integral to be approximated is \(\int_{0}^{2} f(x) dx\), where the function \(f(x)\) is the given pdf.

Step 3: Applying Simpson's Rule

Applying Simpson’s rule to the problem at hand with \(n=6\) sub-intervals. The approximation therefore becomes: \[P(0 ≤ x ≤ 2) ≈ (1/6) ∑_{i=1}^{6} [f(x_{i-1}) + 4f((x_{i-1} + x_{i}) / 2) + f(x_{i})]\] where \(x_{i}\) are points equally spaced from 0 to 2. Calculate each term in the sum and then sum them up.

Step 4: Computing the Final Solution

Upon calculating each term and adding them up, you will get the approximation of the probability that \(x\) is in the interval [0, 2] under the given standard normal distribution.

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special type of normal distribution, sometimes referred to as the Z-distribution. It has a mean of 0 and a standard deviation of 1. This particular distribution is crucial in statistics because it provides a model for how real-world measurements fall around a central point.

Some key points about the standard normal distribution are:

• A symmetric bell-shaped curve centered around the mean.
• About 68% of values fall within one standard deviation of the mean.
• Approximately 95% of values fall within two standard deviations.
• The total area under the curve sums to 1, representing a probability of 100%.

Understanding this distribution helps us assess probabilities and make inferences about larger data sets. The symmetry and characteristics are handy when solving statistical problems, such as finding confidence intervals and conducting hypothesis testing.

Probability Density Function

A probability density function (PDF) is central to understanding how probabilities are distributed across different values in a continuous random variable. For the standard normal distribution, the PDF is given by:
\[ f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \]

This function describes the likelihood of a random variable taking on a specific value. The curve formed by this function is smooth and continuous, representing probabilities at every point along the axis.

Some important properties of PDFs include:

• The total area under the PDF curve equals 1.
• Probabilities for a range of values are computed as the area under the curve between those values.
• PDF cannot give the probability for a single exact value in a continuous distribution; it must be over an interval.

Understanding the PDF is essential in calculating probabilities using methods such as integration, especially when dealing with continuous data.

Numerical Integration

Numerical integration is a technique to approximate the value of integrals, especially when they cannot be solved analytically. Simpson's Rule is one such method used for this purpose.

Simpson's Rule leverages the idea of approximating the function with a series of parabolic arcs between intervals and is expressed as:
\[\int_{a}^{b} f(x)\, dx \approx \frac{b - a}{6}\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\]

Here’s why it's useful:

• It offers better accuracy than simpler methods like the trapezoidal rule.
• Works well with functions that are reasonably smooth and continuous over the interval.
• Particularly effective when used with a small number of sub-intervals.

By applying Simpson’s Rule, we can efficiently calculate the probability that a random variable falls within a specified interval, especially when dealing with complex probability density functions like the standard normal distribution.