Question

In statistics, the standard normal density function is given by \(f(x)=\frac{1}{\sqrt{2 \pi}} \cdot \exp \left[-\frac{x^{2}}{2}\right]\) This function can be transformed to describe any general normal distribution with mean, \(\mu,\) and standard deviation, \(\sigma .\) A general normal density function is given by \(f(x)=\frac{1}{\sqrt{2 \pi} \cdot \sigma} \cdot \exp \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] .\) Describe the transformations needed to get from the graph of the standard normal function to the graph of a general normal function.

Short answer

Shift horizontally by \(\mu\) and scale horizontally by \(\sigma\), then vertically by \(\frac{1}{\sigma}\).

Step-by-step solution

Identify the Standard Normal Density Function

The standard normal density function is given by \[ f(x)=\frac{1}{\sqrt{2 \pi}} \cdot \exp \left[-\frac{x^{2}}{2}\right] \] This function represents a normal distribution with a mean (\(\mu\)) of 0 and standard deviation (\(\sigma\)) of 1.

Understand the General Normal Density Function

The general normal density function is given by \[ f(x)=\frac{1}{\sqrt{2 \pi} \cdot \sigma} \cdot \exp \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] \] where \(\mu\) is the mean and \(\sigma\) is the standard deviation of the distribution. This function allows for any mean and standard deviation, unlike the standard normal density function.

Transformation for the Mean (\(\mu\))

To transform the standard normal function to have a mean of \(\mu\), each point on the graph is shifted horizontally by \(\mu\) units. This changes the function from \[ f(x)= \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[-\frac{x^{2}}{2}\right] \] to \[ f(x)= \frac{1}{\sqrt{2 \pi}} \cdot \exp \left[-\frac{(x - \mu)^{2}}{2}\right] \].

Transformation for the Standard Deviation (\(\sigma\))

The standard deviation changes the spread of the distribution. To adjust the spread, the transformation scales the graph horizontally by a factor of \(\sigma\) and vertically by a factor of \(\frac{1}{\sigma}\). Thus, the function becomes \[ f(x)=\frac{1}{\sqrt{2 \pi} \cdot \sigma} \cdot \exp \left[-\frac{(x - \mu)^{2}}{2 \sigma^{2}}\right] \].

Combine the Transformations

Combining both transformations (mean and standard deviation), the final form of the general normal density function is: \[ f(x)=\frac{1}{\sqrt{2 \pi} \cdot \sigma} \cdot \exp \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] \].

Key concepts

standard normal distribution

The standard normal distribution, often referred to as the Z-distribution, is a fundamental concept in statistics. This distribution has a mean (\text{\textmu}) of 0 and a standard deviation (\text{\textsigma}) of 1. The probability density function for the standard normal distribution is given by: \[ f(x)=\frac{1}{\sqrt{2 \text{\textpi}}} \cdot \text{exp} \left[-\frac{x^{2}}{2}\right] \]
This means that the bulk of the data lies within a standard range, forming the familiar bell-shaped curve centered around 0. The area under this curve equals 1, representing 100% of the data. The standard normal distribution is particularly useful because we can use it as a reference to understand other normal distributions by applying transformations.

mean transformation

To transform a standard normal distribution to a general normal distribution with a different mean (\text{\textmu}), you simply shift the entire graph horizontally by \text{\textmu} units. This results in a new function: \[ f(x)= \frac{1}{\sqrt{2 \text{\textpi}}} \cdot \text{exp} \left[-\frac{(x-\text{\textmu})^{2}}{2}\right] \]
Here, every point on the graph of the standard normal distribution is moved \text{\textmu} units to the right if \text{\textmu} is positive and \text{\textmu} units to the left if \text{\textmu} is negative. This shift changes the center of the distribution but does not affect the shape or spread of the curve.

standard deviation transformation

Adjusting the standard deviation (\text{\textsigma}) of a standard normal distribution impacts the spread of the data. A higher standard deviation means more spread out data, while a lower standard deviation means the data is more clustered around the mean. The transformation for standard deviation modifies the function to: \[ f(x)=\frac{1}{\sqrt{2 \text{\textpi}} \text{\textsigma}} \cdot \text{exp} \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] \]
In this transformation:

• The horizontal scaling factor is \text{\textsigma}, making each unit on the x-axis represent a larger or smaller range of data.
• The vertical scaling factor is \frac{1}{\text{\textsigma}}, ensuring the total area under the curve remains equal to 1.

Combining this with the mean transformation gives the final form of the general normal distribution: \[ f(x)=\frac{1}{\sqrt{2 \text{\textpi}} \text{\textsigma}} \cdot \text{exp} \left[-\frac{(x-\text{\textmu})^{2}}{2 \text{\textsigma}^{2}}\right] \]