Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the normal probability density function $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}} $$ and show, using calculus, that \(\sigma\) is the distance from the mean \(\mu\) to the \(x\) -coordinate of one of the inflection points.

Step-by-Step Solution

Answer

The inflection points are \( x = \mu \pm \sigma \), each \( \sigma \) from the mean \( \mu \).

Step 1: Understand the Normal Distribution Function

The normal probability density function is given by:\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]which represents a symmetric bell-shaped curve centered at the mean \( \mu \) with spread determined by \( \sigma \), the standard deviation.

Step 2: Find the First Derivative

To find the inflection points, we first need the first derivative of \( f(x) \). Using the chain rule and product rule, the derivative is:\[ f'(x) = \left(-\frac{x-\mu}{\sigma^2}\right) \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

Step 3: Set Second Derivative Equal to Zero

Next, we find the second derivative \( f''(x) \), necessary to determine the inflection points. Differentiate \( f'(x) \) using the product rule and simplify:\[ f''(x) = \frac{1}{\sigma^3 \sqrt{2\pi}}\left(\frac{(x-\mu)^2}{\sigma^2} - 1\right) e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]Set \( f''(x) = 0 \) to find inflection points:\[ \frac{(x-\mu)^2}{\sigma^2} = 1 \]

Step 4: Solve for Inflection Points

From \( \frac{(x-\mu)^2}{\sigma^2} = 1 \), we solve for \( x \):\[ (x-\mu)^2 = \sigma^2 \]This gives two solutions: \( x - \mu = \sigma \) or \( x - \mu = -\sigma \), resulting in inflection points at \( x = \mu + \sigma \) and \( x = \mu - \sigma \).

Step 5: Determine Distance from Mean to Inflection Points

The distance between the mean \( \mu \) and each inflection point \( x = \mu + \sigma \) and \( x = \mu - \sigma \) is \( \sigma \). Thus the \( x \)-coordinate of one of the inflection points is a distance of \( \sigma \) from the mean.

Key Concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in statistics, especially when discussing the normal distribution. It is a function that describes the likelihood of a random variable to take on a particular value. In the context of the normal distribution, the PDF is given by:

• The formula: \( f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} \)
• Symmetry around the mean \( \mu \).
• The spread is determined by \( \sigma \), known as the standard deviation.

The normal PDF creates a bell-shaped curve which is symmetric. Most data points are concentrated around the mean, with the probabilities fading out as you move away from it. This curve describes many natural phenomena, such as heights, test scores, and measurement errors, making it a vital tool for data analysis.

Inflection Points

Inflection points are points on a curve where the curvature changes direction. In simpler terms, an inflection point is where the curve shifts from concave (curved upwards) to convex (curved downwards), or vice versa. For the normal distribution curve:

• Inflection points provide insight into the rate of change of the slope.
• They occur at \( x = \mu + \sigma \) and \( x = \mu - \sigma \).

These points are essential because they show where the steepest slope occurs on a normal distribution curve, marking where the change in slope diminishes and then increases again. At these points, the curve passes through its maximum slope, which provides critical information about dispersion in the data.

Calculus Derivatives

Calculus derivatives are tools used to analyze how a function changes. For the normal distribution, derivatives help in finding critical points, including maximums, minimums, and inflection points. The process involves:

• Finding the first derivative: It indicates how the function increases or decreases.
• Calculating the second derivative: It helps determine the concavity of the function.
• Setting the second derivative to zero to find inflection points.

By examining derivatives, we can understand the behavior of the PDF for the normal distribution. The role of derivatives in understanding graphs cannot be overstated as they unveil where functions are curving and help identify points of interest like inflection points efficiently.

Standard Deviation

Standard deviation is a key concept when dealing with normal distribution. It is a measure of the amount of variation or dispersion in a set of values. For the normal distribution:

• \( \sigma \) determines the width of the bell-shaped curve.
• A smaller \( \sigma \) means data is tightly clustered around the mean.
• A larger \( \sigma \) indicates data is spread out over a wider range.

Standard deviation not only helps describe the dispersion of data but also aids in calculating probabilities and understanding the normal distribution curve deeply. It's the distance between the mean and the inflection points on the curve, signifying the spread of the data around the mean.