Lecture 2: Poisson Stascs Probability Density Funcons Expectaon and Variance Es*mators

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Kenneth Gibbs
5 years ago
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1 Lecture 2: Poisso Sta*s*cs Probability Desity Fuc*os Expecta*o ad Variace Es*mators

2 Biomial Distribu*o: P (k successes i attempts) =! k!( k)! p k s( p s ) k prob of each success

3 Poisso Distributio Note that, sice the expectatio value for the umber of successes i oe Beroulli trial is p, the expectatio after summig over idetical Beroulli trials is: =p Now cosider the case where the expected umber of successes depeds o the size of a cotiuous variable (e.g. legth or time iterval), which ca be arbitrarily small. The umber of successes expected over a cotiuous iterval of fiite size ca thus be viewed as resultig from the sum of a ifiite umber of Beroulli trials carried out for arbitrarily small itervals such that: = lim p

4 So, set p= μ/ ad evaluate P (k) = lim! k k!( k)! k = k k! lim! ( k)! k k lim! ( k)! k = lim ( )( 2)...( k)( k )...() ( k)( k )...() k = lim = lim = ( )( 2)...( k + ) k 2 k +...

5 So, set p= μ/ ad evaluate P (k) = lim! k k!( k)! k = k k! lim! ( k)! k k lim h = lim exp log h = lim exp =exp h log i i i

6 So, set p= μ/ ad evaluate P (k) = lim! k k!( k)! k = k k! lim! ( k)! k k lim k =

7 So, set p= μ/ ad evaluate P (k) = lim! k k!( k)! k = k k! lim! ( k)! k k = k e k! Poisso Distributio Coutig statistics, decay processes Iteractio legths cotiuous variable is time cotiuous variable is distace

8 Radioactive Decay: τ = average time for a decay to occur (mea lifetime) = average # decays i time t = t/τ Probability for o decays (=0) withi time t P 0 = ( μ e μ! ) e t/τ P decay = e t/τ (itegral probability) Differetial Probability: P (t) = τ e t/τ Note that this is ow a probability for a cotiuous variable!

9 Poisso distributio: the probability of success depeds o cotiuous variable, but the observatio is a discreet umber of successes. But observatios are ot always of a discreet variable. For cotiuous radom variables (i.e. time, legth, etc.), the probability of obtaiig a particular exact value is geerally vaishigly small (o phase space!). But the relative probability of gettig a value i this viciity versus that viciity is meaigful That s whe you talk about probability desities. But the terms probability distributio ad probability desity fuctio are sometimes iformally used iterchageably.

10 Probability Desity Fuctio (Co*uous radom variables) where Differetial probability as a fuctio of parameters q such that (relative frequecy) q

11 Expectatio Value or for discrete case average value of the fuctio weighted by the frequecy of the depedet variable(s)

12 Is the expectatio the most likely value of the fuctio?.0 NO! 0.5 It does t eve have to be a allowed value of the fuctio! Agle Betwee Shoe Toes ad Isteps The peak is the most likely (frequet) value

13 Is the expectatio the value see half the time? Solar System Plaet Masses Relative to Earth (bis of uit) NO! That s the Media (50% poit) These are oly the same for symmetric distributios

14 The expectatio value tells you how to gamble! How much will you wi, o average, if you play this game very may times?

15 Variace: Average Squared Deviatio from Mea ote: for Poisso: h 2 i = = e X = e " = X =0 2! e = e X = apple ( ) ( )! + ( )! 2 X =2 2 ( 2)! + X = # ( )! ( )! = e " X =2 ( 2)! + X = # ( )! = e 2 (e )+(e )

16 variace = 2 = x 2 2 Uits of σ are same as uits of x (or μ) But, for Poisso, 2 = How do uits work? Here, μ refers to the expected umber of successes, which is uit-less (special case)

17 = p h(x ) 2 i = p hx 2 i 2 = RMS (Root Mea Squared) deviatio uiversal Stadard deviatio whe iterpreted i the cotext of a Normal (Gaussia) distributio

18 Some Useful Cosequeces: The stadard deviatio o a measured umber of couts due to statistical fluctuatios is the square root of the expected mea umber of couts (sqrt of the measured umber is ofte ot a bad approximatio) The expected sesitivity for detectig a sigal i a coutig experimet i terms of the umber of stadard deviatios above backgroud fluctuatios is S/ B I a coutig experimet, the umber of sigal ad backgroud evets detected is proportioal to the coutig time. Thus, the sigal sesitivity goes like T

19 Estimators Ofte we do t kow the true mea ad variace of a distributio ad wat to estimate it from the data: ' X i= x i We wat this to be ubiased, such that the expectatio value is equal to the true value h i = * X i= x i + = X hx i i = () = i= fair eough!

20 2 ' X (x i x) 2 But what about?! i=

21 2 ' X (x i x) 2 But what about?! i= = X * X + X 2 h 2i = (xi x) x 2 i xj * + = X x 2 2 i x X i x j + X X 2 x j x k i * j j k + = X 2 2 X x2 i x i x j + X 2 x 2 j + X X 2 x j x k i 2 j6=i j j k6=j 3 = X 4 2 x 2 2 X i hx i x j i + X x 2 X X 2 j + 2 hx j x k i5 i j6=i j j k6=j = X apple 2 ( ) ( )2 + 2 ( )+ ( )2 2 i = 2 Biased!! So istead, take 2 ' X (x i x) 2 i= Quick Argumet: For every data poits, there are - idepedet measures of the variace (thus correctig the offedig factor)!

22 Variace i the Estimated Mea Note that: var(αx) = (αx) 2 αx 2 = α 2 ( x 2 x 2 ) = α 2 var(x) So, cosider: σ 2 m = var ( x i i= ) = var x 2 ( i i= ) = 2 i= var (x i ) For idepedet variables (as will be show i lecture 4) = 2 ( σ 2 ) = σ2 or σ m = σ

Approximations and more PMFs and PDFs

Approximations and more PMFs and PDFs Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.