READING 17: DISTRIBUTIONS

Uniform Distribution

        A continuous uniform distribution is one where the probability of X occurring in a possible range is the length of the range relative to the total of all possible values. Letting a and b be the lower and upper limit of the uniform distribution, respectively, then for: a ≤ x1 ≤ x2 ≤ b,

Binomial Distribution

        The binomial distribution is a discrete probability distribution for a random variable, X, that has one of two possible outcomes, success or failure. The probability of a specific number of successes in n independent binomial trials is:

        where p = the probability of success in a given trial

Poisson Distribution

        The Poisson random variable X refers to a specific number of successes per unit. The probability for obtaining X successes, given a Poisson distribution with parameter λ is:

Normal Distribution

        The normal probability distribution has the following characteristics:

• The normal curve is symmetrical and bell-shaped with a single peak at the exact center of the distribution.
  
• Mean = median = mode, and all are in the exact center of the distribution.
  
• The normal distribution can be completely defined by its mean and standard deviation because the skew is always zero and kurtosis is always three.

Lognormal Distribution

        A lognormal distribution exists for random variable Y, when Y = e^X, and X is normally distributed.

Student’s t-Distribution

        The t-distribution is similar, but not identical, to the normal distribution in shape—it is defined by the degrees of freedom, has a lower peak, and has fatter tails. The t-distribution is used to construct confidence intervals for the population mean when the population variance is not known. Degrees of freedom for the t-distribution is equal to n − 1; Student’s t-distribution is closer to the normal distribution when df is greater, and confidence intervals are narrower when df is greater.

Chi-squared Distribution

        The chi-squared distribution is asymmetrical, bounded below by zero, and approaches the normal distribution in shape as the degrees of freedom increase.

F-Distribution

        The F-distribution is right-skewed and is truncated at zero on the left-hand side. The shape of the F-distribution is determined by two separate degrees of freedom.

The Central Limit Theorem

        The central limit theorem states that for a population with a mean μ and a finite variance σ2, the sampling distribution of the sample mean of all possible samples of size n will be approximately normally distributed with a mean equal to μ and a variance equal to σ2/n.
        When a sample size is large, the sums of independent and identically distributed (i.i.d.) random variables will be normally distributed.

Mixture Distribution

        Mixture distributions combine the concepts of parametric and nonparametric distributions. The component distributions used as inputs are parametric while the weights of each distribution within the mixture are based on historical data, which is nonparametric.