Question

Investigate the probabilities of an "outlier" for a contaminated normal random variable and a normal random variable. Specifically, determine the probability of observing the event \(\\{|X| \geq 2\\}\) for the following random variables (use the \(\mathrm{R}\) function pcn for the contaminated normals): (a) \(X\) has a standard normal distribution. (b) \(X\) has a contaminated normal distribution with cdf \((3.4 .15)\), where \(\epsilon=0.15\) and \(\sigma_{c}=10\). (c) \(X\) has a contaminated normal distribution with cdf \((3.4 .15)\), where \(\epsilon=0.15\) and \(\sigma_{c}=20\). (d) \(X\) has a contaminated normal distribution with cdf \((3.4 .15)\), where \(\epsilon=0.25\) and \(\sigma_{c}=20\).

Short answer

The probabilities of an outlier for a standard normal distribution and varying contaminated normal distributions can be found using the formulas provided. For specific values, R must be used for the actual calculations.

Step-by-step solution

Standard normal distribution calculation

Calculate the probability for a standard normal distribution using the formula 1 - 2*(1 - Phi(2)), where Phi is the cumulative distribution function (CDF). Using R programming functions, this would be written as 1 - 2*(1 - pnorm(2)).

Contaminated normal distribution calculation - first case

Calculate the probability for the contaminated normal distribution with parameters epsilon=0.15, sigma_c=10. Use the R function pcn with the arguments c(3.4,.15) and parameter equivalent to sigma_c=10. Subtract this value from 1 to get the probability for |X| >= 2: 1 - pcn(2, c(3.4,.15), 10).

Contaminated normal distribution calculation - second case

Repeat the calculation for the contaminated normal distribution with parameters epsilon=0.15, sigma_c=20: 1 - pcn(2, c(3.4,.15), 20).

Contaminated normal distribution calculation - third case

Repeat the calculation for the contaminated normal distribution, but this time with parameters epsilon=0.25, sigma_c=20: 1 - pcn(2, c(3.4,.15), 20, 0.25).

Key concepts

Outlier Probabilities

When dealing with statistical data, it is crucial to understand the concept of 'outlier probabilities'. An outlier is an observation that is significantly different from the rest of the data. It can be caused by variability in the measurement or it may indicate experimental error. The probabilities of outliers are important in statistics because they can significantly affect the results of an analysis.

In the given exercise, the problem is focused on the probability that a random variable, denoted as \(X\), falls outside of a specified range, specifically \(|X| \geq 2\). Determining these probabilities helps us understand the distribution and characteristics of the data. For a standard normal distribution, outliers are less likely because the data is more concentrated around the mean, while contaminated normal distributions can have a higher probability of outliers due to the ’contamination’ which makes the distribution deviate from normality.

Standard Normal Distribution

A standard normal distribution, also known as a Z distribution, is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is the classic bell-shaped curve where the total area under the curve is 1. The further away from the mean an observation lies, the lower the probability of it occurring.

In our exercise, part (a) addresses this distribution. We calculate the probability of an event where \(X\) is greater than or equal to 2, which is quite far from the mean in a standard normal distribution. Mathematically, this probability is \(P(|X| \geq 2) = 1 - 2 \times (1 - \Phi(2))\), where \(\Phi\) is the cumulative distribution function representing the area under the curve to the left of the given value. In simpler terms, it’s the likelihood that a value is less than or equal to the one we've chosen to investigate.

Cumulative Distribution Function (CDF)

In probability theory and statistics, a cumulative distribution function (CDF) maps the probability that a real-valued random variable \(X\) with a given probability distribution will be found at a value less than or equal to \(x\). Essentially, it tells us the probability that a random variable is less than a certain value.

For a standard normal distribution, the CDF, denoted as \(\Phi(x)\), is symmetric around zero and has a sigmoid shape. It approaches zero as \(x\) goes to negative infinity and approaches one as \(x\) goes to positive infinity. In the case of contaminated normal distributions, the CDF can take on different shapes, depending on the degree and type of contamination, as seen with the different parameter values for epsilon and sigma_c in parts (b), (c), and (d) of our exercise.

R Programming

R is a programming language and software environment used for statistical analysis, graphics representation, and reporting. For this exercise, we utilize R's capability to solve statistical problems with built-in functions like pnorm, which calculates the cumulative probability for a standard normal distribution, and pcn, a hypothetical function provided for the purpose of calculating probabilities for a contaminated normal distribution.

The use of programming in statistics offers precision and efficiency, allowing us to compute complex probabilities quickly and with high accuracy. For example, the command pnorm(2) in R calculates the probability that a standard normally distributed variable is less than 2. By utilizing these functions, students can effectively analyze and understand different distributions and their properties.