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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