Features and Applications of various Probability Distributions

Jeremiah Vestors Obiero

D33/ 2154/ 04

Loss Models

BMS 407

Mr. Wekesa

20 November 2008

1.Features and Applications of Binomial Distribution

For a distribution to be described as having consistent with a binomial model, it must fulfill the following features:

An outcome on each trial of an experiment is classified into one of the two mutually exclusive categories: successive or failure

The random variable counts the number of successes in a fixed number of trials

The probability of success remains the same for each trial and so does the probability for failure

The trials in a binomial distribution are independent and therefore the outcome of one trial does not affect the outcome of any other trial

1.1Applications of Binomial Distribution

The binomial distribution is used to analyze the error in experimental results that estimate the proportion of individuals in a population that satisfy a condition of interest.

When a coin is flipped, the outcome is either a head or a tail; when a magician guesses the card selected from a deck, the magician can either be correct or incorrect, when a baby is born, the baby is either born in the month of May or not. In each of these examples, an event has two mutually exclusive possible outcomes

If an event occurs N times (for example, a coin is flipped N times), then the binomial distribution can be used to determine the probability of obtaining exactly r successes in the N outcomes. The binomial probability for obtaining r successes in N trials is given by:

2.Features and Applications of a Normal Distribution

A normal distribution has the following characteristic features:

It is bell-shaped and has a single peak at the center of the distribution

Arithmetic mean, median and mode of the distribution are equal and located at the peak

It is symmetrical about its mean [i.e. if it is cut vertically at the central value the two halves will be mirror images].

The distribution is asymptotic; this is to mean that the curves get closer and closer to the x- axis but they do not touch

The area under the total curve is equal to 1 (A= 1)

In its application, a normal distribution is applied in many areas in life. Although some statistical methods, such as the t test, are not sensitive to moderate departures from normality, it is generally preferable not to rely on this feature. Visual inspection of the distribution may suggest whether the assumption of normality is reasonable but, as figure 3 suggests, this approach is unreliable. Significance tests and normal plots can be used to assess formally whether sample data are a plausible sample from a normal population.1 When data do not have a normal distribution we can either transform the data (for example, by taking logarithms) or use a method that does not require the data to be normally distributed. We consider these topics in future notes.

The normal distribution has another essential place in statistics. Just as separate samples selected at random from the same population will differ (fig 3), so will calculate statistics such as the mean blood pressure. We can think of the means from many samples as themselves also having a distribution. A key theoretical result, called the central limit theorem, underpins many methods of analysis. It states that the means of random samples from any distribution will themselves have a normal distribution. As a consequence, when we have samples of hundreds of observations we can often ignore the distribution of the data. Nevertheless, because most clinical studies are of a modest size, it is usually advisable to transform non-normal data, especially when they have a skewed distribution.

We can consider binary attributes in the same way. For example, the proportions of individuals with asthma will vary from sample to sample. If having asthma is represented by the value 1 and not having asthma by the value 0 then the mean of these values in the sample is the proportion of individuals with asthma. Thus a proportion is also a mean and will follow a normal distribution. These methods are not valid in small samples–some “exact” methods can be used.2 Similar comments apply to some other statistics, such as regression coefficients or standardized mortality ratios, but for mortality ratios the sample size may have to be very large indeed.

One of the most important applications of these results is in calculating confidence intervals. The general method is based on the idea that the statistic of interest (such as the difference between two means or proportions) would have a normal distribution in repeated samples.3

3.Features and Applications of Poisson Distribution

A Poisson distribution has the following salient features:

The mean and variance of a Poisson distribution are equal

Events in a Poisson distribution occur [i.e. occur one at a time] and at random in a given interval or space

The mean number of occurrences in the given interval is known and finite

The probability of success is usually small and the number of trials is usually large

The application of Poisson distribution has been successful when an extraordinarily large number of natural and social phenomena have been successfully modeled using the Poisson distribution.

The “domain” in which counts are observed can be in interval of time: as for the radioactive counts mentioned above or as for cases arriving at the emergency room of a hospital during a one-hour period in mid-afternoon.

The domain can be a volume. In the volume represented by a beaker containing cells in suspension, the number of cells that divide in a particular unit of time may be modeled with the Poisson distribution. Also the number of red giant stars in a volume of interstellar space has been shown to be Poisson distributed.

The domain can be linear. The number of defects in a length of wire and the number of armadillos killed by traffic on a length of an Arizona highway have both been shown to have Poisson distributions.

The domain can be an area. Bomb hits in acre tracts of metropolitan London during WW2, the number of pollen grains collected in regions of a sticky plate exposed to the open air, and (under the right conditions) the number of bird nests in tracts San Francisco Bay marshes have all been successfully modeled as Poisson.

4.Features and Applications of Bernoulli Distributions

The Bernoulli distribution is a discrete distribution having two possible outcomes labeled by INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline1.gif” * MERGEFORMATINET and INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline2.gif” * MERGEFORMATINET in which INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline3.gif” * MERGEFORMATINET (“success”) occurs with probability INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline4.gif” * MERGEFORMATINET and INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline5.gif” * MERGEFORMATINET (“failure”) occurs with probability INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline6.gif” * MERGEFORMATINET , where INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline7.gif” * MERGEFORMATINET . It therefore has probability function

The performance of a fixed number of trials with fixed probability of success on each trial is known as a Bernoulli trial.

The distribution of heads and tails in coin tossing is an example of a Bernoulli distribution with INCLUDEPICTURE “http://mathworld.wolfram.com/images/equations/BernoulliDistribution/Inline8.gif” * MERGEFORMATINET . The Bernoulli distribution is the simplest discrete distribution, and it is the building block for other more complicated discrete distributions.

5.Features and Applications of Chi-square (X2) Distribution

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student’s t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables divided by their respective degrees of freedom. This distribution is applied in graduation of mortality rates where it provides useful comparison of the experience and the standard table in the form of single statistics.

6.Features and Application of Student-T Distribution

Student’s distribution arises when (as in nearly all practical statistical work) the population standard deviation is unknown and has to be estimated from the data. It is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. It is the basis of the popular Student’s t-tests for the statistical significance of the difference between two sample means, and for confidence intervals for the difference between two population means.

The t-distribution is used in hypothesis testing where confidence intervals and hypothesis tests rely on Student’s t-distribution to cope with uncertainty resulting from estimating the standard deviation from a sample, whereas if the population standard deviation were known, a normal distribution would be used.

It is also applied in robust parametric modeling of heavy tailed data which the normal distribution does not allow for.

7.Features and Applications of F-Distribution

The F-distribution is a continuous probability distribution. It arises frequently as the null distribution of a test statistic, especially in likelihood-ratio tests, perhaps most notably in the analysis of variance. An F-test therefore is any statistical test in which the test statistic has an F-distribution if the null hypothesis is true. Examples include:

The hypothesis that the means of multiple normally distributed populations, all having the same standard deviation, are equal. This is perhaps the most well-known of hypotheses tested by means of an F-test, and the simplest problem in the analysis of variance (ANOVA).

The hypothesis that a proposed regression model fits well

The hypothesis that the standard deviations of two normally distributed populations are equal, and thus that they are of comparable origin.

Bibliography

Consul Prem C. Famoye Felix and, Kotz, Samuel (2005) “Applications of Probability Distributions in Real Situation”

Wasserman, L. (2006) “All of Nonparametric Statistics” Springer Publications