Question

Suppose the pdf \(f(x)\) is symmetric about 0 with cdf \(F(x)\). Show that the probability of a potential outlier from this distribution is \(2 F\left(4 q_{1}\right)\), where \(F^{-1}(0.25)=\) \(q_{1}\) Use this to obtain the probability that an observation is a potential outlier for the following distributions. (a) The underlying distribution is normal. Use the \(N(0,1)\) distribution. (b) The underlying distribution is logistic; that is, the pdf is given by $$ f(x)=\frac{e^{-x}}{\left(1+e^{-x}\right)^{2}}, \quad-\infty

Short answer

The probability of an observation being a potential outlier can be calculated using the formula \(2F(4q_{1})\) for each of the given probability distributions: Normal, Logistic and Laplace, where \(q_{1}\) (the first quartile) for each distribution is calculated by equating the cumulative distribution function (CDF) to 0.25 and solving for \(x\)

Step-by-step solution

Calculation for a Normal Distribution

For a standard normal distribution \(N(0,1)\), the first quartile \(q_{1}\) is the value of \(x\) for which \(F(x) = 0.25\). This value obtained from standard normal distribution tables is approximately -0.674. Substitute \(q_{1}\) into the given formula to obtain the probability that an observation from a normal distribution is a potential outlier: \(2F(4\times-0.674)\). Refer to the standard normal distribution table for \(F(x)\) at \(x = -2.697\) to find the probability.

Calculation for a Logistic Distribution

For a logistic distribution, the pdf \(f(x)=\frac{e^{-x}}{(1+e^{-x})^{2}}\). To find the first quartile \(q_{1}\) we solve the equation \(F(x) = 0.25\) for x where \(F(x)\) is the CDF of the logistic distribution: \(F(x) =\frac{1}{1+ e^{-x}}\). After finding the value of \(q_{1}\), similarly substitute \(q_{1}\) into the given formula to calculate the probability of a potential outlier.

Calculation for a Laplace Distribution

The pdf for a Laplace Distribution is given as: \(f(x) = \frac{1}{2} e^{-|x|}\). We can determine the CDF for two cases: when \(x > 0\), \(F(x) = 1 - \frac{1}{2}e^{-x}\) and when \(x < 0\), \(F(x) = \frac{1}{2} e^{x}\). To find \(q_{1}\), we'll have to solve for \(x\) in the equation \(F(q_{1}) = 0.25\). After finding the first quartile \(q_{1}\), we can plug it into the outlier formula to calculate the probability of a potential outlier.

Key concepts

Understanding Probability Density Function (PDF)

The Probability Density Function (PDF) is a fundamental concept in the field of mathematical statistics. It describes the likelihood of a continuous random variable to take on a particular value. The PDF is denoted as \(f(x)\), where \(x\) represents a continuous variable. The key characteristic of the PDF is that the area under the curve between two points represents the probability that the variable falls within that interval. For instance, the PDFs mentioned in the exercise for logistic and Laplace distributions are peculiar types of functions that distinctly characterize the distributions.

For the logistic distribution, the PDF formula given by \(f(x)=\frac{e^{-x}}{(1+e^{-x})^{2}}\), depicts that as \(x\) moves away from zero, the probability decreases following a logistic curve. On the other hand, the Laplace distribution with its PDF \(f(x)=\frac{1}{2}e^{-|x|}\), suggests a steeper decline, reflecting how values far from zero are less likely in a Laplace distribution compared to a logistic distribution. To visualize these, imagine a graph where the y-axis represents the probability density and the x-axis the variable values; the graph of the PDF will show you how the probabilities are distributed across different values of \(x\).

In practice, this information is crucial. For instance, in quality control processes, knowing the PDF allows for the prediction of the occurrence of certain measurements. Understanding the shape and spread of the PDF is also valuable in identifying the central tendency and variability of the data, ensuring better decision-making based on probabilistic outcomes.

Cumulative Distribution Function (CDF) Explained

Moving on to the Cumulative Distribution Function (CDF), it's essentially a 'running total' of probabilities. Given a PDF, the CDF is obtained by integrating the PDF from negative infinity to any value \(x\), denoted by \(F(x)\). It indicates the probability that a random variable is less than or equal to \(x\). A CDF will always range from 0 to 1 and is non-decreasing; each point on the CDF graph signifies the total probability up to that point.

The exercise highlights how one can utilize the CDF in determining the probability of an event. For example, the quartile \(q_1\) which is the value below which 25% of the observations can be found, is deduced from the CDF. To find the probability of a potential outlier, specifically in symmetric distributions about zero, you can use the CDF to calculate \(2F(4q_1)\), where \(q_{1}\) is the first quartile. It reflects the probability of having values at the extreme ends of the distribution, beyond a certain threshold.

Applying CDF in Real-life Scenarios

Understanding how to use the CDF is not only academically important but also has practical implications, such as in risk assessment where you might want to know the probability of a loss exceeding a certain value, or in meteorology when predicting the chance of rainfall amount exceeding a certain threshold.

Outlier Probability in Distributions

Finally, 'outlier probability' is a term that resonates with anyone who deals with data. In statistics, outliers are data points that are significantly different from the other data points in a dataset. They could either represent a variability in the measurement or indicate experimental errors. The probability of a potential outlier is a measure of how likely it is for an observation to be an extreme value.

The formula for outlier probability provided in the exercise, \(2F(4q_{1})\), is particularly useful for symmetric distributions centered around zero. We generally define outliers based on a certain distance from the central tendency of the data. For symmetric distributions, this method uses the concept of quartiles and multiplies the first quartile (which is on the negative side for symmetric distributions about zero) by four, then looks up this value in the CDF to find the extreme tail probabilities.

Implications of Outlier Probability

An understanding of outlier probability can be immensely beneficial in fields like finance for detecting fraud or in engineering for spotting faulty equipment. It's a tool that can flag anomalies which might otherwise go unnoticed. For students, grasping the concept of outlier probability is key to understanding data distribution and variabilities within different contexts.