Notes02 Davis Chp02 ElementaryStatistics - CLAS...
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1 Davis Chp. 2 – Elementary Statistics Probability (pp. 1124) Binomial Combinations Number of combinations of n items taken r at a time: ! ! = !! ! − ! ! !! Binomial Distribution – The classic coin toss example: r successes will occur in n trials ! = ! ! (1 − !) !!! ! ! = !! ! − ! ! !! (1 − !) !!! ! ! Negative Binomial – “How many wells do we have to drill before a r discoveries are made?” ! = (! + ! + 1)! ! − 1 ! !! (1 − !) ! ! ! Sampling with/without Replacement Hypergeometric Probability Distribution – “What are the chances of x discoveries out of n drill holes if N prospects contain S reservoirs?” ! = ! ! !!! !!! ! ! = !! ! − ! ! !! (! − !)! (! − !) − (! − !) ! (! − !)! !! ! − ! ! !! Mutually Exclusive Events and Additive Rule of Probability When only a discrete number of outcomes are possible and they are all mutually exclusive, then ! ! !" ! !" ! = ! ! + ! ! + !(!) Independent Events and the Multiplicative Rule of Probability Conditional Probabilities If harmonic tremors occur from magma movement in a volcano AND eruptions follow magma movement in a volcano, then there’s a relationship between harmonic tremors and eruptions, so the probability of a tremor AND an eruption occurring is not equal to the probability of a tremor X the probability of an eruption. Bayes’ Theorem – The joint probability that both events A and B occur is equal to the probability that B will occur given that A has already occurred times the probability that A will occur. Bayes’ Basic Equation:
Transcript of Notes02 Davis Chp02 ElementaryStatistics - CLAS...
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Davis -‐ Chp. 2 – Elementary Statistics
Probability (pp. 11-‐24)
Binomial Combinations -‐ Number of combinations of n items taken r at a time:
!! =
!!! − ! ! !!
Binomial Distribution – The classic coin toss example: r successes will occur in n trials
! =!! (1− !)!!!!! =
!!! − ! ! !! (1− !)
!!!!!
Negative Binomial – “How many wells do we have to drill before a r discoveries are made?”
! =(! + ! + 1)!! − 1 ! !! (1− !)
!!!
Sampling with/without Replacement
Hypergeometric Probability Distribution – “What are the chances of x discoveries out of n drill holes if N prospects contain S reservoirs?”
! =!!
!!!!!!!!
=
!!! − ! ! !!
(! − !)!(! − !)− (! − !) ! (! − !)!
!!! − ! !!!
Mutually Exclusive Events and Additive Rule of Probability
When only a discrete number of outcomes are possible and they are all mutually exclusive, then
! ! !" ! !" ! = ! ! + ! ! + !(!)
Independent Events and the Multiplicative Rule of Probability
Conditional Probabilities -‐ If harmonic tremors occur from magma movement in a volcano AND eruptions follow magma movement in a volcano, then there’s a relationship between harmonic tremors and eruptions, so the probability of a tremor AND an eruption occurring is not equal to the probability of a tremor X the probability of an eruption.
Bayes’ Theorem – The joint probability that both events A and B occur is equal to the probability that B will occur given that A has already occurred times the probability that A will occur.
Bayes’ Basic Equation:
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!(!,!) = ! !|! !(!)
Converse of Bayes’ Basic Equation:
!(!,!) = ! !|! !(!)
Equating Bayes’ Basic Equation and its converse:
! !|! !(!) = ! !|! !(!)
Solving for the probability of B’s occurrence, given A’s prior occurrence gives Bayes’ Theorem in its common form:
! !|! =! !|! !(!)
!(!)
!(!) = ! !|!! !(!!)!
!!!
Bayes’ Theorem for individual events that are conditionally related to A:
! !!|! =! !|!! !(!!)! !|!! !(!!)!
!!!
Nice Example illustrating the use of Bayes’ Theorem provided – determination of likely source area of a fossil found downstream of a tributary.
Continuous Random Variables (pp. 25-‐29)
Increasing number of coin flips narrows the bin widths on the histogram, decreasing the probability that any one result (e.g. 21 out of 50 flips will be heads) will occur.
Experimental Error and Confounded Sources of Variation
Normal Distribution
Population and Sample
Statistics (pp. 29-‐33)
Parameters describe characteristics of population distributions, whereas Statistics describe characteristics of sample distributions.
Frequency Histogram vs. Relative Frequency Histogram
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Cumulative Plot
Quantiles, Percentiles, Deciles, Quartiles
Box-‐and-‐Whisker Plots – show population distributions in a cartoon-‐like fashion
Summary Statistics (pp. 33-‐39)
