Gamma Distribution Applications(See Wikipedia)

• The gamma distribution has been used to model the aggregate size of insurance claims.

• The gamma distribution has been used to model the aggregate amount of rainfall accumulated in a reservoir.

• The gamma distribution is used in Bayesian statistics.

The Gamma distribution

α is the shape parameter and β the scale parameter.

Gamma density curves

Exponential Distribution Applications

1. The Exponential Distribution is often useful in situations involving waiting times.

2. Where the number of occurrences of a phenomenon follows a Poisson distribution, the time between occurrences follows an exponential distribution.

3. Some situations involving an exponential random variable:1. The time it takes for your next text message to arrive.2. Distance between mutations on a DNA strand.3. Time between calls to a help desk.4. Time between arson reports in a city.5. Lifetimes.

The exponential distribution

Exponential density curves

Standard Beta Distribution 3

Beta is frequently used to model probabilities and proportions. Bayesians frequently use Beta to model prior distributions. Look at the following problem and its R solution:

Standard Beta Distribution

The standard Beta distribution with parameters α and β has the density function:

If X has a standard Beta distribution then X only takes on values in [0,1]. X will have expected value

And variance:

Chi-Squared Distribution facts

• The chi-squared distribution (also chi-square or χ²-distribution) with k degrees of freedom is the distribution of the the sum of k independent standard normal random variables.

• Used is testing for goodness of fit, comparing counts in tables and independence.

• A chi-squared distribution with ν degrees of freedom has expected value ν and variance 2ν.

The Chi-squared distribution

Chi-squared densities

Student’s T distribution-ApplicationsFrom Wikipedia

Student’s T distribution

Student’s T distribution

Weibull Distribution Applications• Some models for which the Weibull distribution has

proven valuable include:– Weather forecasting (Wind speed distributions)– Insurance Claims(Reinsurance) and cumulative development

of asbestosis losses– Hydrology (Maximum 1-day rainfalls)– Describing the size of particles generating by grinding,

crushing, milling

• The reference for the above is the Weibull distribution entry from Wikipedia

Weibull Distribution

α is the shape parameter and β is the scale parameter.

The cdf is given by:

expected value variance

Weibull Distribution

By varying α and β we gave get many density function shapes from the Weibull family.

Very often, by varying α and β we can find a Weibull density that fits our data fairly closely.

Example

Suppose the time for carrion flies to find and begin laying eggs in a fresh exposed corpse has a gamma distribution with shape parameter = 8 min and scale parameter = 150 min.

On average, how long would you expect to wait for flies to start showing up?

What is the probably that flies start showing up between 60 and 120 minutes?

Example

The probability density function between waiting times for successive (independent) events to occur is exponential with rate l.

Malicious probing for open ports on a networked computer occurs at a rate of about 10/hour. What is the probability that you will be probed in the next 10 minutes?

What is the probability that your computer can go a whole day without being violated by a hacker?

Example

The probability of a transfer for fibers from a an item of clothing to a location where that item had been is variable, but believed to follow a Beta distribution with parameters α=7 and β=10.5.

On average, what is the probability that a sweater owned by a suspect transferred fibers found in the trunk of a car where a body was found?

What is the probability that the fibers were transferred with probability between 51% and 80%?

Example

The “Mahalanobis distance” between the RI and density of a known fragment of glass recovered on a suspects clothing and a fragment of glass recovered from a shooting victim is 3.418. This statistic follows a chi-squared distribution with two degrees of freedom.

What is the probability of obtaining a value of this distance at least this great from a randomly chosen pair of glass fragments?

What is the probability of obtaining a value for this statistic from two randomly chosen fragments between values of 1.4 and 5.4?

Example

A pill found on a suspect is weighed four times. The masses are: 9.7987g 9.6609g 10.3882g 9.8717g.

What is the average mass and its standard deviation?

What are the standardized scores and the average score?

What is the probability that a value this extreme or less under the t-distribution.

Example

After studying several similar locations, it was found that the amount of time (in months) small dust bunnies linger before moving on (i.e. persist at a location) follows a Weibull distribution with shape and scale parameters of approximately 1.5 and 1

What is the probability that a dust bunny deposited when a crime took place will still be there at least three months later?