Problem 84 - Calculus Single Variable

Question

If \(\mu\) is any real number, and \(\sigma\) is any positive real number, then $$ f_{\mu, \sigma}(x)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right),-\infty

Step-by-Step Solution

Answer

Perform a substitution to show that the probability simplifies to the integral of the standard normal distribution.

Step 1: Understanding the Problem

We need to show that the probability that the Gaussian random variable, centered at \( \mu \) with standard deviation \( \sigma \), falls within the interval \( \mu-k\sigma \leq X \leq \mu+k\sigma \) is equal to the integral of the standard normal distribution from \(-k\) to \(k\).

Step 2: Setting up the Probability

The probability we are trying to prove is:\[ P(\mu-k\sigma \leq X \leq \mu+k\sigma) = \int_{\mu-k\sigma}^{\mu+k\sigma} f_{\mu, \sigma}(x) \, dx \]Using the Gaussian probability density function, we have:\[ \int_{\mu-k\sigma}^{\mu+k\sigma} \frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right) dx \]

Step 3: Substitute & Simplify

To simplify the integral, perform the substitution \( z = \frac{x-\mu}{\sigma} \), hence, \( dx = \sigma \, dz \). This transforms the limits from \( x = \mu-k\sigma \) and \( x = \mu+k\sigma \) to \( z = -k \) and \( z = k \), respectively.

Step 4: Changing the Integral Limits

After substitution, the integral becomes:\[ \int_{-k}^{k} \frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} z^{2}\right) \sigma \, dz = \frac{1}{\sqrt{2 \pi}} \int_{-k}^{k} \exp \left(-\frac{1}{2} z^{2}\right) \, dz \]The \( \sigma \) terms cancel out as planned during substitution transformation.

Step 5: Final Result

Thus, the integral simplifies to:\[ \frac{1}{\sqrt{2 \pi}} \int_{-k}^{k} \exp \left(-\frac{1}{2} x^{2}\right) dx \]This confirms the probability expressed as the integral of the standard normal distribution over the interval from \(-k\) to \(k\).

Key Concepts

Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is one of the most important concepts in statistics and probability theory. It is characterized by the familiar bell-shaped curve, which represents how data points are distributed across a range of values. This distribution is crucial because it appears in various natural phenomena, including measurement errors and natural variations.

• The function that describes this distribution is called the Gaussian function.
• This distribution is defined by the probability density function: \[ f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp\left(-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2\right)\]

Here, \(\mu\) represents the mean, or the center of the distribution, and \(\sigma\) is the standard deviation, dictating the spread or width of the curve around the mean. The properties of the Gaussian distribution make it extremely useful for statistical inference. It is symmetrical around the mean, and it tails off symmetrically as you move away from the mean.

Standard Deviation

Standard deviation is a vital statistical measure that indicates how much variation or dispersion exists from the mean (average) of a dataset. When dealing with the Gaussian distribution, the standard deviation plays a crucial role in determining the shape and spread of the curve.

• If the standard deviation is small, the data points are tightly clustered around the mean, resulting in a sharp peak in the curve.
• If the standard deviation is large, the data is spread out more, and the curve becomes wider and flatter.

Mathematically, the standard deviation \(\sigma\) is the square root of the variance:\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \]In this formula, \(x_i\) represents each data point and \(\mu\) is the mean of the dataset. Understanding standard deviation helps in assessing the reliability and predictability of data observations within a Gaussian distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the Gaussian distribution where the mean \(\mu\) is 0 and the standard deviation \(\sigma\) is 1. This standard form is widely used in statistical analysis and is denoted by the letter \(Z\).

• Its probability density function is simplified to:\[ \phi(z) = \frac{1}{\sqrt{2 \pi}} \exp\left(-\frac{1}{2} z^2\right) \]
• It serves as the foundation for many statistical tests and methods, making it immensely important in hypothesis testing and confidence interval estimation.

A key benefit of the standard normal distribution is that it provides a way to compare different data sets on the same scale. Observations can easily be converted to \(z\)-scores, which measure how many standard deviations a data point is from the mean. This standardization process is essential for comparison and analysis across different normal distributions.

Random Variable

A random variable is a fundamental concept in probability theory used to describe outcomes of a random phenomenon. In the context of Gaussian distribution, the random variable \(X\) follows a Gaussian (normal) distribution characterized by its mean \(\mu\) and standard deviation \(\sigma\).

• Random variables can be discrete, taking on distinct values like whole numbers, or continuous, capable of taking an infinite number of values within a given range, like any value along a real number line.
• Gaussian distributed random variables are continuous, where every outcome within the specified range is possible and follows the distribution's curve.

Probability density functions (PDFs) describe the likelihood of different outcomes for a continuous random variable. For a Gaussian random variable, the PDF outlines the probabilities associated with each possible value of \(X\), which visually corresponds to the Gaussian bell curve. Understanding random variables allows us to model uncertainties and predict the likelihood of various outcomes in probabilistic scenarios.