Question

Computer plot on the same axes the normal probability density functions with \(\mu=0\), \(\sigma=1,\) and with \(\mu=3, \sigma=1\) to note that they are identical except for a translation.

Short answer

Plot the PDFs for \( \mu = 0 \text{ and } \mu = 3 \) and note they are the same shape but translated.

Step-by-step solution

- Define the normal probability density function

The normal probability density function (PDF) is given by the formula: \[ f(x | \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) \]where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

- Set the parameters for the first PDF

For the first normal PDF, use the parameters \( \mu = 0 \) and \( \sigma = 1 \). Thus, the formula becomes: \[ f(x | 0, 1) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{x^2}{2}\right) \]

- Set the parameters for the second PDF

For the second normal PDF, use the parameters \( \mu = 3 \) and \( \sigma = 1 \). Thus, the formula becomes: \[ f(x | 3, 1) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{(x - 3)^2}{2}\right) \]

- Plot the PDFs on the same axes

To compare the two functions, plot \( f(x | 0, 1) \) and \( f(x | 3, 1) \) on the same set of axes. Use graphing software or a plotting library such as Matplotlib in Python. The x-axis should cover a range that includes both centers of the distributions. For example, use an x-axis range from -5 to 8 and ensure that both PDFs are clearly visible.

- Note the translation

Observe that the two curves are identical in shape but one is shifted by 3 units to the right compared to the other. This translation is due to the difference in their means (\( \mu = 0 \) and \( \mu = 3 \)). Both distributions have the same standard deviation (\( \sigma = 1 \)) which means they have the same spread.

Key concepts

Mean Shift

In probability and statistics, the concept of 'mean' can be thought of as the average or central value of a distribution. A 'mean shift' occurs when this central value changes. Imagine you have two identical normal distributions, but one of them is centered at \( \mu = 0 \) and the other at \( \mu = 3 \). The shape of the two distributions is the same, but the entire curve of the second distribution is shifted 3 units to the right on the x-axis. This is precisely what a mean shift refers to: moving the central point of the distribution without affecting its shape. Observing this in a graph helps students understand that while the mean alters the position, it doesn't affect the overall form of the probability distribution function (PDF). A classic example to visualize is plotting two normal PDFs, one with a mean of 0 and the other with a mean of 3 but both with a standard deviation of 1.

Standard Deviation

Standard deviation, symbolized by \( \sigma \), measures the dispersion or spread of a set of values around the mean. In the context of normal distributions, a standard deviation defines the width of the 'bell curve'. The larger the \( \sigma \), the flatter and wider the curve; the smaller the \( \sigma \), the steeper and narrower the curve. When comparing two normal distributions, even if they have different means, having the same standard deviation means they spread out to the same extent. This makes the shapes identical but positioned differently when plotted. In our exercise, both PDFs have \( \sigma = 1 \). Hence, they share the same bell curve shape, only differing by their mean values.

Plotting Distributions

Visualizing data through plots is an essential part of understanding distributions. For plotting normal distributions:

• Choose an appropriate range for the x-axis to include centers of both distributions.
• Use graphing software or plotting libraries like Matplotlib in Python.
• Label the axes and plot different PDFs on the same set of axes for comparison.

Here, to compare the PDFs \( f(x | 0, 1) \) and \( f(x | 3, 1) \):

• Set the x-axis from -5 to 8 (to cover a broad enough range).
• Plot both distributions to see that the only difference is a horizontal shift.
• Ensure both curves are clearly visible and properly labeled.

This visual representation solidifies the understanding of mean shifts and standard deviations.

Comparison of PDFs

Comparing probability density functions (PDFs) helps highlight differences and similarities in distributions. Here are some points to consider:

• Shape: PDFs with the same standard deviation would have identical shapes.
• Position: Different means result in horizontal shifts of the curves.
• Overlap: For distributions with means close to each other and similar standard deviations, there can be significant overlap, indicating similarity in data spread.

In practice: Plotting \( f(x | 0, 1) \) versus \( f(x | 3, 1) \), it's evident they look the same except for one being shifted by 3 units to the right. This comparison visually enforces the concept of mean shift while confirming that the standard deviation remains constant across both distributions.