Question

An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-a x^{2}}\) a. Graph the Gaussian for \(a=0.5,1,\) and 2. b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a). c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.

Short answer

Short answer: For the Gaussian function \(f(x) = e^{-ax^2}\), we graphed the function for different values of a, computed the area under the curve using the formula \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}}\), and evaluated the integral of the generalized Gaussian function \(f(x) = e^{-(ax^2+bx+c)}\) using the method of completing the square, resulting in \(e^{-k}\sqrt{\frac{\pi}{a}}\), where k is determined from completing the square.

Step-by-step solution

Graph the Gaussian function for given values of a

Create a graph of \(f(x)=e^{-ax^2}\) for \(a=0.5, 1\), and 2, either by using a graphing calculator or plotting software like Desmos or GeoGebra. Make sure to label the x and y-axis, as well as differentiating the curves with corresponding values of a.

Compute the area under the curve using the given formula

The formula given for the area under the curve of the Gaussian function is \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}}\). For each value of a, calculate this area using the formula: For \(a=0.5\): \(\sqrt{\frac{\pi}{0.5}}= \sqrt{2\pi}\) For \(a=1\): \(\sqrt{\frac{\pi}{1}}= \sqrt{\pi}\) For \(a=2\): \(\sqrt{\frac{\pi}{2}}= \sqrt{\frac{\pi}{2}}\) These are the areas under the curves for each value of a.

Evaluate the integral of the generalized function using the method of completing the square

To evaluate \(\int_{-\infty}^{\infty} e^{-(ax^2+bx+c)} dx\), we will first complete the square for the quadratic expression in the exponent. Consider the quadratic term \(ax^2 + bx + c\). To complete the square, we will rewrite this term in the form \(a(x-h)^2 + k\). First, factor out the a from the first two terms: \(a(x^2 + \frac{b}{a}x)\). Now, complete the square inside the parentheses: \(a[(x + \frac{b}{2a})^2 - \frac{b^2}{4a^2}]\). So, we have \(a(x-h)^2 + k\) where \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\). Now, rewrite the integral with the completed square: \(\int_{-\infty}^{\infty} e^{-[a(x-h)^2+k]} dx\) Use the substitution: \(u = x-h\), then \(x = u+h\) and \(dx = du\) The integral becomes: \(\int_{-\infty}^{\infty} e^{-[a(u)^2+k]} du\) Use the property of exponentials to separate the terms: \(\int_{-\infty}^{\infty} e^{-au^2} e^{-k} du\) The constant \(e^{-k}\) can be factored out of the integral: \(e^{-k} \int_{-\infty}^{\infty} e^{-au^2} du\) Now, use the given formula \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}}\) to evaluate the integral: \(e^{-k} \sqrt{\frac{\pi}{a}}\) The integral of the generalized Gaussian function is \(e^{-\left(a\left(x-\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}\right)}=\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x=e^{-k}\sqrt{\frac{\pi}{a}}\)

Key concepts

Understanding the Normal Distribution

The normal distribution, also known as the Gaussian distribution, is crucial in statistics and the natural sciences. Visually represented by the iconic bell-shaped curve, the normal distribution is symmetric around the mean and describes how values of a variable are dispersed or spread. The function that represents it is given by the equation \(f(x) = e^{-ax^2}\), where \(a\) determines the width of the bell.

Important properties of the normal distribution include its mean and standard deviation, which tell us the center and the spread of data respectively. Virtually all values lie within three standard deviations from the mean, encapsulating the famous empirical rule, depicting that about 68%, 95%, and 99.7% of values fall within one, two, and three standard deviations.

The normal distribution is omnipresent in statistical analysis, as it approximates a variety of natural phenomena such as heights, test scores, or measurement errors. Hence, understanding it allows interpretation of data, prediction of outcomes, and conducting of hypothesis testing.

Completing the Square in Algebra

The method of completing the square is a powerful algebraic tool often used to solve quadratic equations and evaluate integrals involving a squared term. This method converts a quadratic expression into a perfect square plus a constant. The general form of completing the square transforms \(ax^2 + bx + c\) into \(a(x-h)^2 + k\), where \(h\) and \(k\) represent specific numeric values.

To complete the square:

• Factor out \(a\) from the \(x^2\) and \(x\) terms.
• Add and subtract the square of half the coefficient of \(x\), within the bracket.
• Rewrite the expression as a square of a binomial plus a constant.

Through this process, you produce an equation that is much easier to work with, especially in calculus when finding areas under curves or solving integrals. Completing the square plays a critical role in evaluating the integral of the Gaussian function, transforming it into a more manageable form.

The Integral of Gaussian Functions

The integral of the Gaussian function not only is fundamental in probability theory but also underpins various fields in engineering and physical sciences. The equation associated with this integral, \(\int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}}\), may seem daunting at first glance due to the limits approaching infinity. However, it’s precisely this property that makes the Gaussian integral so useful—enabling the calculation of areas under the entire normal distribution curve.

The process involves completing the square and then using a substitution to simplify the integral. By doing so, the variable we integrate over becomes 'detached' from the rest of the terms, allowing a known result to be used for the actual integration step. Hence, the Gaussian integral gives us a concise representation of the total probability for a normal distribution, essential for statistical analysis and probability theory. Understanding how to evaluate this integral is not just useful for theoretical knowledge but also practically important for real-world applications.