distribution

Examples of distribution in the following topics:

• t Distribution Table

  • T distribution table for 31-100, 150, 200, 300, 400, 500, and infinity.

• Student Learning Outcomes

• The Normal Distribution

  • Normal distributions are a family of distributions all having the same general shape.
  • The normal distribution is a continuous probability distribution, defined by the formula:
  • If $\mu = 0$ and $\sigma = 1$, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate.
  • The simplest case of normal distribution, known as the Standard Normal Distribution, has expected value zero and variance one.
  • Many sampling distributions based on a large $N$ can be approximated by the normal distribution even though the population distribution itself is not normal.

• Chi Square Distribution

  • Define the Chi Square distribution in terms of squared normal deviates
  • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
  • The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2, since 4 is 22.
  • As the degrees of freedom increases, the Chi Square distribution approaches a normal distribution.
  • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.

• The t-Distribution

  • Student's $t$-distribution (or simply the $t$-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
  • The $t$-distribution with $n − 1$ degrees of freedom is the sampling distribution of the $t$-value when the samples consist of independent identically distributed observations from a normally distributed population.
  • The $t$-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth.
  • Fisher, who called the distribution "Student's distribution" and referred to the value as $t$.
  • Note that the $t$-distribution becomes closer to the normal distribution as $\nu$ increases.

• Common Discrete Probability Distribution Functions

  • Your instructor will let you know if he or she wishes to cover these distributions.
  • A probability distribution function is a pattern.
  • These distributions are tools to make solving probability problems easier.
  • Each distribution has its own special characteristics.
  • Learning the characteristics enables you to distinguish among the different distributions.

• The Standard Normal Distribution

  • The standard normal distribution is a normal distribution of standardized values called z-scores.
  • For example, if the mean of a normal distribution is 5 and the standard deviation is 2, the value 11 is 3 standard deviations above (or to the right of) the mean.
  • The mean for the standard normal distribution is 0 and the standard deviation is 1.
  • The transformation z = (x − µ)/σ produces the distribution Z ∼ N ( 0,1 ) .
  • The value x comes from a normal distribution with mean µ and standard deviation σ.

• Shapes of Sampling Distributions

  • The "shape of a distribution" refers to the shape of a probability distribution.
  • The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central; and where types of departure from this include:
  • A normal distribution is usually regarded as having short tails, while a Pareto distribution has long tails.
  • Even in the relatively simple case of a mounded distribution, the distribution may be skewed to the left or skewed to the right (with symmetric corresponding to no skew).
  • Sample distributions, when the sampling statistic is the mean, are generally expected to display a normal distribution.

• Sampling Distributions and the Central Limit Theorem

  • The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution.
  • The normal distribution has the same mean as the original distribution and a variance that equals the original variance divided by $n$, the sample size.
  • For large enough $n$, the distribution of $S_n$ is close to the normal distribution with mean $\mu$ and variance
  • The upshot is that the sampling distribution of the mean approaches a normal distribution as $n$, the sample size, increases.
  • The usefulness of the theorem is that the sampling distribution approaches normality regardless of the shape of the population distribution.

• History of the Normal Distribution

  • The importance of the normal curve stems primarily from the fact that the distributions of many natural phenomena are at least approximately normally distributed.
  • Laplace showed that even if a distribution is not normally distributed, the means of repeated samples from the distribution would be very nearly normally distributed, and that the larger the sample size, the closer the distribution of means would be to a normal distribution.
  • Because the distribution of means is very close to normal, these tests work well even if the original distribution is only roughly normal.
  • Examples of binomial distributions.
  • The smooth curve is the normal distribution.