Erlang distribution

(noun)

The distribution of the sum of several independent exponentially distributed variables.

Related Terms

Examples of Erlang distribution in the following topics:

  • The Exponential Distribution

    • The exponential distribution is a family of continuous probability distributions that describe the time between events in a Poisson process.
    • The exponential distribution is a family of continuous probability distributions.
    • Another important property of the exponential distribution is that it is memoryless.
    • The length of a process that can be thought of as a sequence of several independent tasks is better modeled by a variable following the Erlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).
    • Reliability engineering also makes extensive use of the exponential distribution.

  • 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.