Question

Obtain the probability that an observation is a potential outlier for the following distributions. (a) The underlying distribution is normal. (b) The underlying distribution is the 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 an outlier in any distribution needs the integral of the density function beyond a certain point usually defined by several standard deviations from the mean. Values can change based on the specific distribution and defined cutoff values. For example, for the Normal distribution, this corresponds to values outside +/- 3 standard deviations from the mean.

Step-by-step solution

Normal Distribution

By definition, a potential outlier in a normal distribution is a data point that resides three or more standard deviations away from the mean. In a standard normal distribution, this corresponds to \(|z|\) greater than 3. The total probability for observing these values can be found by taking one minus the cumulative density between -3 and 3 for a standard normal distribution. So, \(P(Z > 3 or Z < -3) = 1 - P(-3 < Z < 3) = 1 - (F(3) - F(-3))\) where F is the cumulative distribution function (CDF) of the standard Normal distribution.

Logistic Distribution

For the logistic distribution, the potential outlier is typically defined as lies in the tails of the distribution where the probability density falls low enough. We can follow the same process as with the normal distribution. The computation is slightly more complex due to the form of the distribution: \(P(X > k or X < -k) = 1 - P(-k < X < k) = 1 - (F(k) - F(-k))\) where F is the cumulative distribution function (CDF) of the logistic distribution.

Laplace Distribution

We use the similar approach for Laplace distribution. Let's consider a data point to be an outlier if it is beyond three scale parameters from the mean: \(P(X > 3b or X < -3b) = 1 - P(-3b < X < 3b) = 1 - [F(3b) - F(-3b)]\), where F is the CDF of the Laplace distribution, and b is referred to as 'diversity,' roughly analogy to standard deviation.

Key concepts

Normal Distribution

The normal distribution is one of the most widely used probability distributions in statistics. It is symmetric and bell-shaped, characterized by its mean (average) and standard deviation (spread). For a normal distribution, data follows the 68-95-99.7 rule, meaning approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

When identifying potential outliers in a normal distribution, we're looking for data points lying beyond three standard deviations from the mean. This is because such points are rare (less than 0.3% probability) and appear on the extreme tails of the distribution. To find the probability of an observation being an outlier, calculate the probability of a data point being beyond these limits using the cumulative distribution function (CDF).

In practice, this involves computing:

• \(P(Z > 3 \text{ or } Z < -3) = 1 - (F(3) - F(-3))\)
• The CDF, \(F\), captures the probability that a random variable takes a value less than or equal to \(x\).
• Using statistical software or a standard z-table helps find these probabilities efficiently.

Logistic Distribution

The logistic distribution is similar to the normal distribution but with heavier tails, meaning it has more observations at the extremes. It is defined by a probability density function (pdf), where it calculates the likelihood of different outcomes.

Its CDF is critical in determining outlier probabilities, especially since this distribution is frequently used in logistic regression and the modeling of binary outcomes. It closely approaches 0 and 1 more gradually than the normal distribution, impacting the distribution of outliers.

To find potential outliers, use the logistic distribution's CDF similarly. Outliers are typically beyond a specific range in the tails where the pdf's values are naturally low:

• \(P(X > k \text{ or } X < -k) = 1 - (F(k) - F(-k))\)
• The choice of \(k\) reflects how defined outliers are in the context of this distribution.
• Consideration of how the logistic's gradual tail behavior impacts this will aid in understanding the extremities present in your data.

Laplace Distribution

The Laplace distribution, or double exponential distribution, has a sharp peak at the mean and heavier tails than the normal distribution. It is characterized by its mean and diversity (b), akin to a standard deviation.

Because of these heavy tails, outliers in the Laplace distribution might appear more frequently than in a normal distribution. The CDF is used to find probabilities of outlying observations, which can be seen beyond multiples of its diversity parameter.

To check for potential outliers:

• \(P(X > 3b \text{ or } X < -3b) = 1 - [F(3b) - F(-3b)]\)
• Here, \(b\) is the diversity, which is crucial in determining how far outliers might be and their probabilistic estimate.
• With heavier tails, a higher occurrence of observations at the extremes is expected.

Understanding this helps accurately model situations where extreme values are likely.