The number of phone calls arriving at a call center within a minute Erlang The

The number of phone calls arriving at a call center

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The number of phone calls arriving at a call center within a minute [Erlang] The number of goals in sports involving two competing teams The number of jumps in a stock price in a given time interval Under an assumption of homogeneity, the number of times a web server is accessed per minute The number of mutations in a given stretch of DNA after a certain amount of radiation The proportion of cells that will be infected at a given multiplicity of infection The arrival of photons on a pixel circuit at a given illumination and over a given time period The targeting of V-1 flying bombs on London during World War II The counts of prime numbers in short intervals obey a Poisson distribution provided a certain version of an unproved conjecture of Hardy and Littlewood is true [Gallagher] COS 424/SML 302 Probability and Statistics Review February 6, 2019 58 / 69
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Modeling number of siblings using a Poisson distribution 0.0 0.1 0.2 0.3 0.4 0 2 4 Number of siblings density Histogram of number of siblings The empirical mean number of siblings: ˆ λ = 1 . 3; empirical variance is 1. COS 424/SML 302 Probability and Statistics Review February 6, 2019 59 / 69
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Gaussian density -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 N(1.2, 1) x p(x) The mean μ controls the location of the center of the distribution: x N ( μ, σ 2 ) x - μ N (0 , σ 2 ) COS 424/SML 302 Probability and Statistics Review February 6, 2019 60 / 69
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Gaussian density -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 N(1.2, 1) x p(x) The variance σ 2 controls the spread of the distribution. The standard deviation σ is the square root of variance. p ( μ - σ x μ + σ ) 0 . 6827 p ( μ - 2 σ x μ + 2 σ ) 0 . 9545 p ( μ - 3 σ x μ + 3 σ ) 0 . 9973 COS 424/SML 302 Probability and Statistics Review February 6, 2019 61 / 69
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Important distributions to know: Gaussian Support : x ∈ < Parameters : μ , the mean, and σ 2 , the variance Probability density function : p ( x | μ, σ ) = 1 2 πσ exp - ( x - μ ) 2 2 σ 2 MLE estimates of μ (the empirical mean): ˆ μ = x 1 n n X i =1 x i MLE estimates of σ 2 (the empirical variance): ˆ σ 2 = 1 n n X i =1 ( x i - x ) 2 Conjugate prior for μ : Gaussian distribution Conjugate prior for σ 2 : Inverse gamma distribution COS 424/SML 302 Probability and Statistics Review February 6, 2019 62 / 69
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Data modeled using a Gaussian distribution The central limit theorem (CLT) : under certain conditions, the arithmetic mean of a sufficiently large number of observations of independent random variables will be approximately normally distributed regardless of the underlying distribution COS 424/SML 302 Probability and Statistics Review February 6, 2019 63 / 69
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Data modeled using a Gaussian distribution In biology, the log of random variables tend to have a normal distribution (after separating male/female subpopulations) including: Measures of size (length, height, skin area, weight) The length of inert appendages (hair, claws, nails, teeth) Physiological measurements, such as blood pressure of adult humans In finance, in particular the Black-Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest,
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  • Spring '09
  • Probability theory

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