Question

Using \(\mathrm{R}\), investigate the probabilities of an "outlier" for a \(t\) -random variable and a normal random variable. Specifically, determine the probability of observing the event \(\\{|X| \geq 2\\}\) for the following random variables: (a) \(X\) has a standard normal distribution. (b) \(X\) has a \(t\) -distribution with 1 degree of freedom. (c) \(X\) has a \(t\) -distribution with 3 degrees of freedom. (d) \(X\) has a \(t\) -distribution with 10 degrees of freedom. (e) \(X\) has a \(t\) -distribution with 30 degrees of freedom.

Short answer

The probabilities of an outlier (|X| >= 2) are different for each distribution and degree of freedom. For a standard normal distribution, it is approximately 0.045, for a t-distribution with 1 degree of freedom it is about 0.317, for 3 degrees of freedom it is around 0.103, for 10 degrees of freedom it is approximately 0.059 and for 30 degrees of freedom it is about 0.048.

Step-by-step solution

Probabilities for Standard Normal Distribution

Firstly, solve for a standard normal distribution. A standard normal distribution in \( R \) uses the pnorm function. To find the probability of \( |X| \geq 2 \), find the one-sided probability of getting less than -2 (which is the probability of \( X \leq -2 \)). Then, double this value because \( |X| \geq 2 \) includes both ends of the distribution. Thus, use the command 2 * pnorm(-2).

Probabilities for \( t \)-Distribution with 1 Degree of Freedom

Next, find the probability for \( t \)-distribution using 1 degree of freedom. \( R \) uses the pt function for \( t \)-distribution. Again, double the one-sided value because \( |X| \geq 2 \) includes both sides of the distribution. Use the command 2 * pt(-2, df = 1).

Probabilities for \( t \)-Distribution with 3 Degrees of Freedom

Repeat steps like in step 2, but this time with 3 degrees of freedom. Use the command 2 * pt(-2, df = 3).

Probabilities for \( t \)-Distribution with 10 Degrees of Freedom

For 10 degrees of freedom, adapt the command to 2 * pt(-2, df = 10).

Probabilities for \( t \)-Distribution with 30 Degrees of Freedom

For 30 degrees of freedom, use the command 2 * pt(-2, df = 30).

Key concepts

Standard Normal Distribution

The standard normal distribution, commonly known as the bell curve, is a probability distribution that has a mean of zero and a standard deviation of one. In this context, it helps to understand the likelihood of random variable values falling within specific ranges.

When a question asks for the probability of the event \( |X| \geq 2 \), it is asking for the likelihood that the value of X is at least 2 units away from the mean, either above or below. This is crucial for identifying outliers, which are values significantly different from others in a dataset. Calculating this probability involves symmetry in the bell curve, allowing us to calculate one tail and then double it to account for both. The probability of an 'outlier' is the area under the curve starting from the value 2 to positive infinity and its equivalent area on the negative side of the distribution.

T-Distribution

The t-distribution, or Student's t-distribution, is closely related to the standard normal distribution but slightly differs in its shape. It’s particularly useful when dealing with smaller sample sizes or when the population standard deviation is unknown. Unlike the standard normal distribution, the t-distribution has thicker tails, meaning that it predicts a greater likelihood of outlier values.

For each number of degrees of freedom, the shape of the t-distribution is unique. As the degrees of freedom increase, however, the t-distribution approaches the shape of the standard normal distribution. It's critical to correctly choose the degrees of freedom, often represented by 'df', because they reflect constraints in a dataset, such as the number of samples minus one.

Degrees of Freedom

Degrees of freedom are a concept in statistics that refer to the number of independent values or quantities which can be assigned to a statistical distribution. In the context of the t-distribution, the degrees of freedom usually represent the number of independent data points minus the number of parameters estimated from the data.

These influence the shape of the t-distribution: the lower the degrees of freedom, the fatter the tails, and hence, the higher the probability of finding outliers. It is fundamental to get this number right in probability calculations because it directly affects the outcome. Wrong degrees of freedom can lead to inaccurate conclusions about the likelihood of events or the nature of the dataset involved.

Probability Calculations in R

In R, probability distributions come to life through various functions that enable easy computation of different probabilities. For the standard normal and t-distributions covered here, the pnorm and pt functions are used, respectively.

These functions compute the cumulative probability up to a certain value for the specified distribution. When using these functions, the output provides the probability that a random variable from the corresponding distribution will have a value less than or equal to the specified value. For asymmetrical calculations, like finding the probability of \( |X| \geq 2 \), these functions help determine the probability in one tail, and then the result is doubled to account for the full range of interest. R's capabilities make analyzing complex probability questions an efficient process for statistical practitioners.