Problem 59 - Essential Calculus

Question

Probability In Exercises 59 and \(60,\) approximate the normal probability with an error of less than \(0.0001,\) where the probability is given by \(P(a< x < b)=\frac{1}{\sqrt{2 \pi}} \int_{a}^{b} e^{-x^{2} / 2} d x\). $$ P(0

Step-by-Step Solution

Answer

The approximate probability \( P(0 < x < 1) \) can be calculated using Simpson's rule and evaluating the function \( f(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2} \) at \( x = 0, 0.5, 1 \), then substituting these values into the formula of Simpson's rule.

Step 1: Defining the Function

Define the function \( f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} \). This is the probability density function of the standard normal distribution.

Step 2: Applying the Simpson's Rule

Simpson's rule can be used to solve the integral numerically. We can achieve an error of less than \(0.0001\) using Simpson's rule by setting the step size to be small. Given an integral in the form \( \int_a^b f(x) dx \), Simpsons rule is given by \( \frac{(b-a)}{6} * [f(a) + 4*f((a+b)/2) + f(b)] \). For this problem, \( a = 0, b = 1 \). Now, substitute these values into Simpson's rule formula to compute the approximation of the integral.

Step 3: Calculating The Probability

Substitute \( a = 0, b = 1 \) into Simpson's rule to obtain the approximation of probability \( P \approx \frac{1}{6} [f(0) + 4*f(0.5) + f(1)] \). Evaluate \( f(0), f(0.5), f(1) \) and substitute these values into the formula to obtain the approximation of probability \( P \).

Key Concepts

Normal Distribution

The normal distribution is a fundamental concept in statistics and probability. It describes how the values of a variable are distributed along a symmetrical, bell-shaped curve. This distribution is called 'normal' because it often perfectly fits the natural variability observed in various phenomena, like heights, test scores, or measurement errors.

• A normal distribution is characterized by its mean, which determines where the center of the curve is, and its standard deviation, which indicates the spread or width of the curve.
• The standard normal distribution is a special case where the mean is 0 and the standard deviation is 1.

Many statistical methods rely on the normal distribution due to its convenient mathematical properties. Understanding the shape and features of this curve helps interpret statistical data effectively.

Simpson's Rule

Simpson's Rule is a technique used in numerical integration to approximate the value of a definite integral. It is especially useful when dealing with functions that cannot be easily integrated analytically, like complex probability functions.

• The formula for Simpson's Rule in approximating the integral from \( a \) to \( b \) is \( \frac{b-a}{6} [f(a) + 4f((a+b)/2) + f(b)] \).
• This method works by fitting a quadratic polynomial through three points on the curve and calculating the area under this curve as an approximation of the integral.

For better accuracy, Simpson's Rule can be applied over smaller intervals, achieving results within a predefined error range. This is ideal for computing probabilities, as accurate results are critical in such applications.

Probability Density Function

A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value. For a variable that follows a normal distribution, its PDF provides a way to calculate probabilities of occurrence for specific intervals.

• The PDF formula for the standard normal distribution is \( f(x) = \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} \).
• The area under a PDF curve over a specified interval represents the probability that the variable falls within that interval.

Thus, the PDF is a core concept when determining probabilities for continuous datasets, paving the way to thorough statistical analyses.

Numerical Integration

Numerical integration is a set of techniques for determining the value of an integral when an analytical solution is difficult or impossible to obtain. In many practical situations, especially those involving complex or non-standard functions, numerical methods are required to approximate integrals with acceptable accuracy.

• Methods like Simpson's Rule, the Trapezoidal Rule, and others are commonly used to perform numerical integration.
• These techniques usually involve dividing the area under the curve into small shapes (like trapezoids or curves) and summing their areas.

Using numerical integration allows mathematicians and scientists to handle real-world problems where exact solutions can't be computed, making it an indispensable tool in fields like engineering, computer science, and physics.