Random Numbers Generator and Statistics Set (Set 2)

The normal distribution is the most commonly used probability distribution in statistics. Many other probability distributions are related to this distribution. As the number of random variables increases, the distribution becomes a bell shaped curve. This curve is called the normal curve or Gaussian curve (in honor of the German mathematician Karl Friedrick Gauss, 1777-1855). The normal distribution is defined with mean and standard deviation.

Random Numbers Generator and Statistics Set (Set 2) - 12 Programs

Log Normal Distribution **|** Log Pearson Type III Distribution **|** Normal Distribution

Chi-Square Distribution**|** F-Distribution **|** Student-T
Distribution **|** Multivariate Standard Normal Distribution **|** Gamma Distribution **|** Beta Distribution **|** Hypergeometric Distribution **|** Triangular
Distribution **|** Binomial Distribution

Chi-Square Distribution

The log-normal distribution is often assumed to be the distribution of a stock price. A distribution is log-normally distributed when the natural log of the set of the random variables in that distribution is a normally distributed. In plain English, if you take the natural log of each of the random numbers from a log-normal distribution, the new number set will be normally distribution. Like the normal distribution, log-normal distribtuion is also defined with mean and standard deviation.

The most common use of the chi-square distribution is to test the difference between proportions. It has a positive skew. The skew decreases when degree of freedom increases as the distribution approaches normal. The mean of a chi-square distribution is its degree of freedom.

Student T distribution is used commonly for small sample size -usually a sample size less than 30. A t distribution shares some common characteristics with the standard normal distribution. Both distributions are symmetrical, both range in value from negative infinity to positive infinity, and both have a mean of zero and standard derivation of one. However, a t distribution has a greater dispersion than the standard normal distribution.

The Log Pearson Type III distribution is commonly used in hydraulic studies. It is somehow similar to normal distribution, except instead of two parameters, stanand deviation and mean, it also has skew. When the skew is small, Log Pearson Type III distribution approximates normal.

This program is a derivation of the Multivariate Standard Normal Probability Distribution example. Users will be able to populate random multivariate standard normal deviates on the spreadsheet for analysis. For detail on this distribution, please refer to the Multivariate Standard Normal Probability Distribution example.

The Gamma distribution is most often used to describe the distribution of the amount of time until the nth occurrence of an event in a Poisson process. For example, customer service or machine repair. The Gamma distribution is related to many other distributions. For example, when a Gamma distribution has an alpha of 1, Gamma(1, b), it becomes an Exponential distribution with scale parameter of b, Expo(b). And a Chi-Square distribution with k df is the same as the Gamma(k/2,2) distribution.

The Beta distribution can be used in the absence of data. Possible applications are estimate the proportion of defective items in a shipment or time to complete a task. The Beta distribution has two shape parameters, a1 and a2. When the two parameters are equal, the distribution is symmetrical. For example, when both a1 and a2 are equal to one, the distribution becomes uniform. If a1 is less than a2, the distribution is skewed to the left. And if a1 is greater than a2, the distribution is skewed to the right.

The Hypergeometric distribution is a discrete distribution. It is alike the Binomial distribution. Both of the Hypergeometric distribution and the Binomial distribution describe the number of times an event happens in a fixed number of trials. The difference between the two distributions is that Binomial distribution trials are independent, while Hypergeometric distribution trials change the probability for each subsequent trial and are called sampling without replacement.

The Triangular distribution is often used when no or little data is available. It has 3 parameters, the minimum and the maximum that defines the range, and the more likely (the peak). The distribution is skewed to the left when the peak is closed to the minimum and to the right when the peak is closed to the maximum. It is a simple distribution that as its name implied, has a triangular shape.

The Binomial distribution describes the number of successes in t independent Bernoulli (yes or no) trails with probability p of success on each trial. It is used to answer questions such as how many times a head will come up when a coin is flipped 5 times or how many defective items will be found in 20 items.

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The F distribution is commonly used for ANOVA (analysis of variance), to test whether the variances of two or more populations are equal. For every F deviate, there are two degrees of freedom, one in the numerator and one in the denominator. It is the ratio of the dispersions of the two Chi-Square distributions. As both of the degree of freedom increase, the percentile value is approaching to one. F is also used in tests of ˇ§explained varianceˇ¨ and is referred to as the variance ration, Explained variance/Unexplained variance.

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