Question

Show that the graph of a pdf \(N\left(\mu, \sigma^{2}\right)\) has points of inflection at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).

Short answer

The graph of the probability density function of a normal distribution has inflection points at \(x = \mu + \sigma\) and \(x = \mu - \sigma\).

Step-by-step solution

Recall the pdf of the normal distribution

The probability density function of a normal distribution is given by \[f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^{2}}\].

Find the second derivative of the pdf

The second derivative of \(f(x)\) with respect to \(x\) is \[f''(x) = \frac{1}{\sqrt{2\pi} \sigma^{3}}e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^{2}}\left[\left(\frac{x - \mu}{\sigma}\right)^{2} - 1\right]\]

Solve the equation \(f''(x) = 0\)

Setting \(f''(x)\) equal to zero gives the equation \[\left(\frac{x - \mu}{\sigma}\right)^{2} - 1 = 0\] Solving this equation yields two solutions: \(x = \mu + \sigma\) and \(x = \mu - \sigma\) . These are the points of inflection of the graph of the pdf.

Key concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in statistics that describes the relative likelihood for a random variable to take on a given value. In essence, a PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one specific value.

For continuous variables, the probability of the variable equaling a specific value is zero; thus, the PDF helps to understand the distribution of values around a certain point. The area under the curve of a PDF (between two points) represents the probability that the variable falls within that range.

A high peak on the graph of a PDF indicates that the corresponding value is highly probable, whereas a low point indicates a lower probability. To formally calculate probabilities from a continuous PDF, one would integrate the function over the desired interval.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about its mean, with its shape described as 'bell-shaped'.

The parameters \(\mu\) (mean) and \(\sigma^{2}\) (variance) fully characterize a normal distribution. The mean \(\mu\) determines the center of the distribution, and the standard deviation \(\sigma\) determines its width or 'spread'. A lower standard deviation means the data points are closer to the mean, while a higher standard deviation indicates that the data are more spread out.

In practical terms, a normal distribution is applied to many natural phenomena, from heights and test scores to measurement errors. This distribution becomes extremely useful because it allows statisticians and researchers to make predictions about the population of data.

Second Derivative Test

The second derivative test is a mathematical tool used to determine the concavity of a function and identify points of inflection. A point on the function is a point of inflection if the function changes concavity at that point.

Mathematically, if the second derivative of a function \(f''(x)\) is positive, the graph is concave up (shaped like a cup), and if it's negative, the graph is concave down (shaped like a cap). If the second derivative equals zero and changes sign, it indicates a point of inflection. In the context of the probability density function for a normal distribution, detecting points of inflection assists in understanding where the graph shifts its curvature, which is critical for visualizing and interpreting the behavior of data distributions.

Statistical Concepts

Statistical concepts form the backbone of data analysis and interpretation. They encompass a wide range of topics including measures of central tendency (like mean, median, and mode), variability (standard deviation, variance), and the shape of distributions (skewness, kurtosis).

Understanding these concepts enable us to describe and make inferences about data sets and the relationships between variables. For example, points of inflection in a PDF of a normal distribution are linked to concepts of standard deviation and mean, illustrating how far data points typically deviate from the average.

Moreover, these statistical concepts serve as the foundation for more advanced analyses, such as hypothesis testing, regression, and correlation, which allow for predictions and decision-making based on data.