C.Wright, S. Basant, J.McFarland ESSM 689 02/10/2015

1.  Binomial 2.   Poisson 3.   Normal 4.   Student t 5.   Chi-squared 6.   Others

OUTLINE

Discrete, finite Random sample from a population can be categorized into one of two types: success/failure Assumptions: § Number of trials, n, is fixed § Separate trials are independent § Probability of success is the same for every trial

BINOMIAL DISTRIBUTION

Provides the probability distribution for the number of “successes” in a fixed number of independent trials, when the probability of success is the same in each trial

P[X successes]=( n/X ) p↑x (1−p)↑n−x  X is the number of successes, p is the probability of a success, and n is the number of independent trials

BINOMIAL DISTRIBUTION

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Many trials on the binomial distribution becomes normal…

Discrete Describes the number of successes in time/space Assumptions:

Successes happen independently of each other Equal probability in time/space

POISSON DISTRIBUTION

P[X events]= 𝑒↑−µμ ∗µμ↑𝑋   /𝑋!  Where X is the number of events and

𝜇 is the mean number of events per unit time or space

Good to provide a null hypothesis for testing whether successes occur randomly in time or space

POISSON DISTRIBUTION

Continuous Approximates many phenomena in nature

𝑓(𝑌)= 1/√2πσ    ∗𝑒↑−(𝑌∗µμ)↑2 /2σ  ↑2   

where Y is any real number, 𝜇 is the

mean, and 𝜎 the standard deviation

“bell-shaped”, symmetric about the mean

NORMAL DISTRIBUTION

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NORMAL DISTRIBUTION

About two-thirds of individuals are within one 𝝈 of the 𝝁, and about

95% are within 2𝝈  of the 𝝁 Central Limit Theorem: the sum or mean of a large number of measurements from a non-normal population is approx. normally distributed

The probability distribution of all the values for an estimate that might be obtained when sampling a populations That is, comparing the sample mean to the normal mean.

STUDENT’S T

The t- distribution is given by:

𝒕= 𝒀 𝒔−𝝁  /𝑺𝑬↓𝒀    the difference between the sample mean and the true mean divided by the estimated standard error, with n-1 degrees of freedom

STUDENT’S T

Measures the discrepancy between observed and expected frequencies “goodness of fit” or how good the actual results fit a theoretical distribution model

CHI-SQUARE DISTRIBUTION

with m degrees of freedom:

V= 𝑥↓1↑2 + 𝑥↓2↑2 +…+ 𝑥↓2(𝑚)↑2  Where m is mean, and 2m is variance

The degrees of freedom, or the number of sums, specify which chi-square distribution to use

CHI-SQUARED DISTRIBUTION

Continuous Uniform Distribution Exponential Distribution F Distribution

a)   Describes the arrival time of a randomly recurring independent even sequence

b)   Test whether two population variances are equal; very sensitive to assumption of normal distribution

c)   The probability distribution of random number selection from the continuous interval between a and b. Its density function is defined by:

𝑓(𝑥)={█■1/𝑏−𝑎 &𝑤ℎ𝑒𝑛  𝑎≤x  ≤b  @0&𝑥<𝑎  𝑜𝑟  𝑥>𝑏  

OTHERS