Question

Use Example 6 as a model to show that the graph of the normal probability density function with \(\mu=0\) \(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-x^{2} / 2 \sigma^{2}}\) has points of inflection at \(x=\pm \sigma .\) What is the maximum value of the function? Use a graphing utility to verify your answer by graphing the function for several values of \(\sigma\).

Short answer

The normal probability density function \( f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x^{2} / 2 \sigma^{2})} \) has points of inflection at \(x=\pm \sigma\) and a maximum value of \( f(0) = \frac{1}{\sigma \sqrt{2 \pi}} \).

Step-by-step solution

Find the first derivative

Start by differentiating the given function with respect to \(x\). Use the chain rule for differentiation where necessary. Let \( f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x^{2} / 2 \sigma^{2})} \). In this case, the first derivative \( f'(x) \) can be obtained by applying chain rule, which gives \( f'(x) = -\frac{x}{\sigma^3\sqrt{2\pi}}e^{-(x^2/(2\sigma^2))} \).

Find the second derivative

Taking the derivative of \( f'(x) \) gives the second derivative. Again, use the chain rule in this case, which results in the second derivative \( f''(x) = -\frac{1}{\sigma^3\sqrt{2\pi}}e^{-(x^2/(2\sigma^2))}\left(1-\frac{x^2}{\sigma^2}\right) \).

Find the points of inflection

Points of inflection occur where the second derivative changes sign, i.e., it equals 0 at these points. The equation \( 1-\frac{x^2}{\sigma^2} = 0 \) yields \( x = \pm \sigma \) after solving for \( x \), confirming that \( f(x) \) has inflection points at \( x = \pm \sigma \).

Find the maximum value of the function

The maximum point of the normal distribution curve occurs at its peak. Since \(\mu=0\) in this case, the maximum point on the graph of the function will be at \(x = \mu = 0\). Substituting \(x = 0\) into the original function gives the maximum value, which results in \( f(0) = \frac{1}{\sigma \sqrt{2 \pi}} \).

Key concepts

Probability Density Function

A Probability Density Function (PDF) is at the heart of continuous probability distributions like the normal distribution. It shows how the probabilities are distributed over the values of the random variable. In mathematical terms, for a continuous random variable, the PDF, denoted as \( f(x) \), is a function that describes the likelihood of a random variable to take on a given value.
The normal probability density function for a random variable \( X \) with mean \( \,\mu = 0\) and standard deviation \( \,\sigma \) is expressed as:\[ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x^{2} / 2 \sigma^{2})} \]- The component \( \frac{1}{\sigma \sqrt{2 \pi}} \) ensures the total area under the curve equals 1, which is a requirement for any probability distribution.
- The exponent \( e^{-(x^{2} / 2 \sigma^{2})} \) describes the bell shape of the curve, where smaller standard deviations \( \sigma \) produce taller and narrower curves.
This PDF is central to many statistical concepts due to its predictable properties and is employed in various real-world applications such as calculating probabilities, modeling data, and hypothesis testing.

Points of Inflection

Points of inflection on a graph are where the curvature of the curve changes direction. In simpler terms, it's where the graph changes from being concave (curved upwards) to convex (curved downwards), or vice versa. For the normal distribution, these points can tell us about the shape and spread of the data.
Finding these points involves calculating where the second derivative of the function \( f(x) \) changes sign. The steps are as follows:

• First derivative: Differentiating the normal probability density function gives \( f'(x) = -\frac{x}{\sigma^3\sqrt{2\pi}}e^{-(x^2/(2\sigma^2))} \).
• Second derivative: Further differentiation yields \( f''(x) = -\frac{1}{\sigma^3\sqrt{2\pi}}e^{-(x^2/(2\sigma^2))}(1-\frac{x^2}{\sigma^2}) \).
• Setting \( f''(x) = 0 \) and solving \( 1-\frac{x^2}{\sigma^2} = 0 \) gives the points of inflection as \( x = \pm \sigma \).

At these points, the graph of the probability density function transitions its curvature, providing insights into the distribution's spread surrounding its mean.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes. For the normal distribution, differentiation helps us understand its shape and find important characteristics like the maximum value and points of inflection.
Here's how differentiation is applied to the normal probability density function:

• First derivative \( f'(x) \): This shows the rate of change of the probability density function. Calculated as \( -\frac{x}{\sigma^3\sqrt{2\pi}}e^{-(x^2/(2\sigma^2))} \), it helps find the slope at any point \( x \).
• Second derivative \( f''(x) \): This checks for concavity or convexity. Given by \( -\frac{1}{\sigma^3\sqrt{2\pi}}e^{-(x^2/(2\sigma^2))}(1-\frac{x^2}{\sigma^2}) \), it's used to determine points of inflection and analyze changes in the curve's direction.
  Differentiation reveals critical insights about the behavior of the function, making it easier to understand its geometry and probability characteristics.

Graphing Utility

Graphing utilities are valuable tools for visualizing complex functions like the normal probability density function. They help verify analytical solutions and provide intuitive understanding.
To graph the normal distribution using a graphing utility:

• Input the function \( f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x^{2} / 2 \sigma^{2})} \).
• Adjust the parameter \( \sigma \) to see how changes affect the shape and spread of the curve.
• Analyze key features: the peak at \( x=0 \) and inflection points at \( x=\pm\sigma \).

Using graphing tools provides a concrete visual representation of mathematical conclusions. It confirms calculated points of inflection and maximums and illustrates how the normal distribution's characteristics shift with changes in \( \sigma \). This practical approach bridges theoretical mathematics with real-world interpretation, enhancing comprehension of the normal distribution's properties.