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MATH 2209 Probability Notes Compilation

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MATH 2209: Probability Table of Contents Distributions ............................................................................................................................................................................................. 3 Discrete .......................................................................................................................................................... 3 Binomial X~Bin(n,p) .................................................................
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   MATH 2209: Probability  Callum Biggs 20498066 MATH 2209: Calculus pg. 1   Table of Contents   Distributions ............................................................................................................................................................................................. 3   Discrete .......................................................................................................................................................... 3   Binomial X~Bin(n,p) .............................................................................................................................................................3   Poisson X~Poi( λ ) ..................................................................................................................................................................3   Continuous ..................................................................................................................................................... 4   Uniform U~(a,b) ..................................................................................................................................................................4   Exponential X~Exp( λ ) ...........................................................................................................................................................4   Gaussian X~N( μ,σ 2 ) ..............................................................................................................................................................5   Joint ............................................................................................................................................................................................................. 5   Discrete .......................................................................................................................................................... 5   Discrete Joint PMF ...............................................................................................................................................................5   Finding Discrete Marginal PMF ............................................................................................................................................5   Discrete Joint CDF................................................................................................................................................................5   Stating CDF ..........................................................................................................................................................................5   Continuous ..................................................................................................................................................... 6   Continuous Joint PDF ...........................................................................................................................................................6   Continuous Marginal PDF ....................................................................................................................................................6   Continuous Joint CDF...........................................................................................................................................................6   Proving Joint CDF ............................................................................................................................................ 6   Conditional Probability .................................................................................................................................... 7   Discrete ...............................................................................................................................................................................7   Continuous ..........................................................................................................................................................................7   Multiplication Rule ..............................................................................................................................................................7   Discrete: ..............................................................................................................................................................................7   Continuous: .........................................................................................................................................................................7   Independence ................................................................................................................................................. 7   Discrete Independence ........................................................................................................................................................7   Continuous Independence ...................................................................................................................................................7   Expectancy ................................................................................................................................................................................................ 8   Discrete .......................................................................................................................................................... 8   Continuous ..................................................................................................................................................... 8   Symmetry Theorem ......................................................................................................................................... 8   Covariance and Correlation ............................................................................................................................. 8   Covariance ..........................................................................................................................................................................8   Correlation ..........................................................................................................................................................................9   Independence .....................................................................................................................................................................9   Properties of Covariance and Correlation ......................................................................................................... 9   Properties of Expectation ................................................................................................................................ 9   Functions of Single Variables ............................................................................................................................................................... 9   Definition ........................................................................................................................................................ 9   Determining S Y  ..................................................................................................................................................................10   CDF of functions of Single Variables ............................................................................................................... 10   PDF of functions of Single Variables ............................................................................................................... 10   Process to determine pdf ..................................................................................................................................................11   Corollary: Special Case .......................................................................................................................................................11   Determining PDF via Direct Approach ................................................................................................................................11   E(X) in Single Variable Functions .................................................................................................................... 11   Independence ............................................................................................................................................... 11   Multivariable Functions ..................................................................................................................................................................... 12   Multivariate CDF ........................................................................................................................................... 12   Process of finding multivariate CDF ...................................................................................................................................12   Multivariate PDF ........................................................................................................................................... 12    Callum Biggs 20498066 MATH 2209: Calculus pg. 2   Independence between X and Y ..................................................................................................................... 12   Sums of Two Variables .................................................................................................................................. 13   Moment Generating Functions ......................................................................................................................................................... 13   Definition ...................................................................................................................................................... 13   Finding the MGF ................................................................................................................................................................14   Finding E(  X  n ) ......................................................................................................................................................................14   Tabular ..............................................................................................................................................................................14   Case 1 .......................................................................................................................................................................................................... 14   Case 2 .......................................................................................................................................................................................................... 14   Uniqueness ................................................................................................................................................... 14   Single Variable .............................................................................................................................................. 15   MGF for Functions of Single Variables ...............................................................................................................................15   Corollary  ..................................................................................................................................................................................................... 15   PDF Convolutions ..............................................................................................................................................................15   Summation of Independent Random Variables......................................................................................................................................... 15   Corollary: IID Random Variables ............................................................................................................................................................... 15   Multivariate .................................................................................................................................................. 16    Callum Biggs 20498066 MATH 2209: Calculus pg. 3   Distributions Discrete Discrete Distributions take on finite valuesProbabilities are specified by a pmf          Discrete functions are given in a table form   =    =     ∈   ,   =    =     ∈       =  −    2  =   2 −   2   Binomial X~Bin(n,p)   Models the number of success in a series of independent events, each with a probability of success p Parameters: n = number of trialsp = probability of success     =  (   =  )=    1 − − , ∈ℝ , ≤ 1    =  !  ! − !       =  1 − +          =     =  (1 − )   Poisson X~Poi( λ )   Probability of events occurring in a fixed time period occurs with a known average rate and are independent of the time since the last event  Parameter: Average rate is   For the probability of exactly n occurrences, with an average of   in the time frame     =    −  !       =  −    ! ∈         =exp{   − 1  }      =     =   
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