Problem 77 - Thomas Calculus

Question

Normal probability distribution function The function $$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$ is called the normal probability density function with mean \(\mu\) and standard deviation \(\sigma\) . The number \(\mu\) tells where the distribution is centered, and \(\sigma\) measures the "scatter" around the mean. $$\begin{array}{l}{\text { From the theory of probability, it is known that }} \\\ {\qquad \int_{-\infty}^{\infty} f(x) d x=1}\end{array}$$ In what follows, let \(\mu=0\) and \(\sigma=1\) a. Draw the graph of \(f\) . Find the intervals on which \(f\) is increasing, the intervals on which \(f\) is decreasing, and any local extreme values and where they occur. b. Evaluate $$\int_{-n}^{n} f(x) d x$$ for \(n=1,2,3\) c. Give a convincing argument that $$\int_{-\infty}^{\infty} f(x) d x=1$$ (Hint: Show that \(0 < f(x) < e^{-x / 2}\) for \(x > 1,\) and for \(b > 1\) , $$\int_{b}^{\infty} e^{-x / 2} d x \rightarrow 0 \text { as } b \rightarrow \infty . )$$

Step-by-Step Solution

Answer

a) Graph: It's a bell curve symmetric around 0; increasing on \((-\infty, 0)\), decreasing on \((0, \infty)\). b) Integrals: 0.6827 for \(n=1\), 0.9545 for \(n=2\), 0.9973 for \(n=3\). c) Total area under the curve is 1.

Step 1: Understand the Properties of Normal Distribution

The normal probability distribution function is expressed as \(f(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}x^2}\) when \(\mu = 0\) and \(\sigma = 1\). It is a bell-shaped curve centered at zero, with a total area under the curve equal to 1. The symmetry about the vertical axis at \(x = 0\) is a key feature.

Step 2: Graph the Function

The function \(f(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}x^2}\) is graphed as a bell-shaped curve centered at the origin with symmetry about the y-axis. The peak is at \(x = 0\), indicating the highest probability density and tapering off as \(|x|\) increases.

Step 3: Find Intervals of Increase and Decrease

The function \(f(x)\) is increasing on \((-\infty, 0)\) and decreasing on \((0, \infty)\) due to its symmetry and peak at \(x = 0\). There are no local extremities within \(f(x)\) besides this peak at the origin because it only decreases or changes after this point.

Step 4: Evaluate the Integral for Given Intervals

Using the properties of the normal distribution, evaluate \(\int_{-n}^{n} f(x) dx\). For \(n = 1\), \(\approx 0.6827\), for \(n = 2\), \(\approx 0.9545\), and for \(n = 3\), \(\approx 0.9973\). These integrals represent the probabilities of the random variable falling within \(-1\) to \(1\), \(-2\) to \(2\), and \(-3\) to \(3\), respectively.

Step 5: Show Total Integral Equals One

Using the given hint, compare \(f(x)\) to \(e^{-x/2}\) for large \(x\) and prove that \(\lim_{b \to \infty} \int_b^\infty e^{-x/2} dx = 0\). Therefore, \(f(x)\) covers all possibilities for the probability, justifying \(\int_{-\infty}^\infty f(x) dx = 1\).

Key Concepts

Probability Density Function

A Probability Density Function (PDF) describes how the values of a random variable are distributed. In the context of a normal distribution, the PDF is given by the equation: \[ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \] where \(\mu\) is the mean and \(\sigma\) is the standard deviation. The PDF gives us the likelihood of a random variable being exactly equal to some value \(x\). However, because the probability at any single point on a continuous distribution is theoretical, we often use the area under the curve (integral calculus) to find probabilities over an interval. - The area under the curve from \(-\infty\) to \(\infty\) is always 1. This means the total probability for all possible outcomes is 1, preserving the foundational rule of probability. - The function is defined such that it is always non-negative, and the shape of the bell curve is determined by \(\mu\) and \(\sigma\). A higher \(\sigma\) indicates a wider spread. Understanding the PDF is crucial for predicting the likelihood of different outcomes in a probabilistic model.

Gaussian Distribution

The Gaussian Distribution, also known as the Normal Distribution, is one of the most important probability distributions in statistics. It's characterized by its bell-shaped curve which is symmetric around the mean (\(\mu\)). Here are some fundamental properties: - The distribution is completely determined by its mean \(\mu\) and standard deviation \(\sigma\). - The highest point on the bell curve is located at the mean, \(\mu\). This is where the data is most densely packed, and the likelihood of values near the mean is highest. - Approximately 68% of the data falls within one standard deviation (\(\sigma\)) of the mean, 95% within two standard deviations, and 99.7% within three. This is known as the empirical rule. The shapes of the normal curve change with varying \(\sigma\) values, making it crucial in understanding data variability. Higher \(\sigma\) leads to broader curves, indicating more variability.

Integral Calculus

Integral Calculus is essential in probability for finding the area under probability density curves, such as the normal distribution. This area corresponds to the probability that a random variable falls within a certain range. For the normal distribution, solving integrals helps quantify such probabilities for given ranges: - For instance, to find the probability of a random variable within a specific interval \([-n, n]\), we compute \[ \int_{-n}^{n} f(x) \, dx \]. For the standard normal distribution (\(\mu = 0\), \(\sigma = 1\)), common values like \(n = 1, 2, 3\) correspond to probabilities of 68.27%, 95.45%, and 99.73%, respectively. - Integral calculus is also pivotal in demonstrating that the total area under the curve (the entire set of possible outcomes) equals 1. This ensures that probabilities across the distribution sum to one, which underpins probability theory. To fully leverage the power of the normal distribution, integral calculus enables exact computations of probabilities, a critical step in statistical analyses.