Chapter 7: Special Distributions
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1 This chater first resents some imortant distributions, and then develos the largesamle distribution theory which is crucial in estimation and statistical inference Discrete distributions The Bernoulli distribution X has a Bernoulli distribution with arameter [0, ] if PrX =)= and PrX =0)= Its robability function is Its exectation and variance are x ) x for x =0or 0 otherwise EX) =, VarX) = ), and its mgf is ψt) =Ee Xt )=e t + The binomial distribution X has a binomial distribution with arameters n and if its f is C x n x ) n x for x =0,,,n 0 otherwise This is just the robability that x successes occur in a Bernoulli rocess with n trials The exectation and variance of X are EX) = n VarX) = n ), and its mgf is ψt) =e t + ) n Relationshi with the Bernoulli distribution: if X,,X n are n indeendent Bernoulli random variables with arameter, thenx = P n X i has a binomial distribution with n and
2 The Poisson distribution X has a Poisson distribution with arameter λ>0 if its f is e λ λ x x! for x =0,,, 0 otherwise Both of its mean and variance are λ Itsmgfis ψt) =e λet ) if X,,X n are n indeendent random variables and X i has the Poisson distribution with mean λ i,then nx X = has the Poisson distribution with mean P n λ i the binomial distribution with n and willtendtothepoissondistributionwith mean n when n,,andn is fixed The binomial distribution s f is where λ = n = λx The geometric distribution X i nn ) n x +) x ) n x x! µ n n x + n λ x µ λ n x! n n n n n {z } {z } λ λx e x!, e λ X has a geometric distribution with arameter if its f is ) x for x =0,,, 0 otherwise This is just the robability that x failures has occurred when the first success is realized in a Bernoulli trial, and so X is just the number of failures when the first success is achieved in a Bernoulli trial The mean and variance of X are and its mgf is ψt) = EX) = VarX) =, for t< ln ) )et
3 Continuous Distributions The normal distribution This is the most widely used continuous distribution There are at least two reasons for this: first, the observed distributions of natural henomena are often close to the normal distribution; second, some imortant summaries of a large random samle have the normal distribution aroximately The univariate case X is a normal or Gaussian) distribution with mean μ and variance σ,denotedby X Nμ, σ ),ifitsdfis ex µ ) x μ for <x< πσ σ Its df is which, however, has no closed form F x) = Z x ft)dt It mgf is ψt) =ex µμt + σ t When μ =0and σ =,wecallitastandard nomral distribution We often use the following notation: φx) = ex µ x π and Using this notation, we have and F x) = = Z x Φx) = Z x φt)dt µ x μ σ φ σ Z x μ σ = Φ x μ σ ) ex πσ π ex 3 µ ) t μ dt σ ¾ ½ z dz
4 Thus, all normal variables can be easily reduced to the standard normal by this standardisation Some roerties of the normal distribution: the df of a normal distribution Nμ, σ ) is symmetric with resect to the axis x = μ This means that the mean and median of this distribution coincide in articular, φx) =φ x) and Φx) = Φ x) τ =/σ is called the recision if X Nμ, σ), then where a and b are scalars ax + b Naμ + b, a σ ) if the random variables X,,X n are indeendent and if X i Nμ i,σ i ),then the sum nx nx nx X i N μ i, σ i ) therefore, the samle mean of a random samle with size n from a normal distribution Nμ, σ ) has a normal distribution Nμ, σ /n) Exercise i) Find the 05, 05, 09 quantiles of the standard normal distribution iii) Let X, X and X 3 be indeendent lifetimes of memory chis Suose that each X i has a normal distribution with mean 300 hours and standard deviation 0 hours Comute the robability that at least one of the three chis lasts at least 90 hours The multivariate case We can now define a vector of random variables, X =[X X n ] T We say that this vector follows a n-dimensional normal distribution with mean μ and covariance matrix Σ if the df of this vector is ½ π) n Σ ex ¾ x μ)t Σ x μ) if all random variables are uncorrelated ie, ρ ij =0for i 6= j), then ny fx i ) this also imlies that, whenever normally distributed random variables are uncorrelated, they are also indeendent 4
5 the multivariate standard normal distribution has df φx) = ½ ex ¾ π) n xt x A few results: if X =[X X n ] T follows a multivariate normal distribution then each of its elements is normally distributed in articular, if X,X ) is has a bivariate normal distribution Ã" # " #! μ N σ σ, μ σ σ, then the marginal distribution of X i is normal: X i Nμ i,σ i ); and the conditional distribution of X given X = x is or µ X X = x N X μ σ X = x N where ρ is the correlation coefficient Distributions related to the normal We introduce several concets first: μ +x μ ) σ σ,σ σ σ µ x μ ) ρ, ρ σ A random samle is a list of indeendently and identically distributed iid) random variables Given a random samle {X,X,,X n }, the samle mean is the samle variance is P n X n = X i n P n Sn = X i X n ) n This result can be generalized for higher-dimensional cases see, eg, Greene 008)) 5
6 The Chi-square distribution If {X,,X n } is a random samle from the the standard normal distribution ie, X i N0, )), then nx has a Chi-square distribution with n degrees of freedom denoted by χ n), of which the df is n Γn/) x n e x for x>0 0 otherwise χ is the gama distribution see below) with α = n/ and β =/ E [X] =n, Var[X] =n, and its mgf is Mt) = X i µ n/ for t< t if indeendent random variables X j j =,,k)haveχ distribution with n j degrees of freedom, then P k j= X j has a χ distribution with P k j= n j degrees of freedom Exercise Suose that a oint X, Y, Z) is to be chosen at random in three dimensional sace, where X, Y,andZ are indeendent random variables and each has a standard normal distribution What is the robability that the distance from the origin to the oint will be less than one unit? The t distribution Suose Z N0, ) and Q X nthen X = Z r n Q has a Student s t distribution with n degrees of freedom Its df is for x R µ Γn +)/) nπγn/) + x n n+ for n>, E[X] is zero since fx) is symmetric The mean does not exist when n =) For n>, Var[X] =n/n ) E[ X k ] < for k<nbut E[ X k ]= for k n Nomgfexists 6
7 as n, fx) converges to φx), the standard normal distribution suose {X i },,n form a random samle from a normal distribution with mean μ and variance σ Then n Xn μ) S n has the t distribution with n degrees of freedom Exercise 3 Suose X and X are indeendent and each has a normal distribution with zero mean and variance σ Determine the value of X + X ) Pr X X ) < 4 3 The F distribution Let Y and Z be indeendent variables have χ distributions with m and n degrees of freedom, resectively Then X = Y/m Z/n follows an F distribution with m and n degrees of freedom its df is E[X] =n/n ) if n> Γ [m + n)/] Γm/)Γn/) x m/ mx + n) m+n)/ /X has an F distribution with n and m degrees of freedom if Z has a t distribution with n degrees of freedom, then X = Z has an F distribution with and n degrees of freedom 3 Other continuous distributions The Gamma distribution X has a gamma distribution with arameters α and β bothofthemareositive)ifits df is β α Γα) xα e βx for x>0 0 for x 0, where Γα) = R 0 x α e x dx is the gamma function 7
8 The mean and variance of X are EX) = α β, VarX) = α β, and its mgf is ψt) = µ β α for t<β β t if X,,X n are n indeendent random variables and X i has the Gamma distribution with arameters α i and β, then X = has the Gamma distribution with arameters P n α i and β The exonential distribution The exonential distribution is a secial gamma distribution with α =Itsdfisthus βe βx for x>0 0 for x 0 nx the momoryless roerty of exonential distributions: X i PrX t + h) PrX t + h X t) = PrX t) = e βt+h) e βt = PrX h) To understand this roerty, let us suose that X isthetimethatelasesbefore some event occurs eg, a roduct breaks down) Then this roerty says that, given that the event has not occurred in the first t eriods, then the robability that the event will not take lace in the next h eriods is the same as the robability that the event would not occur during an interval of h eriods starting from the very beginning In other words, regardless of the length of time that has elased without the occurrence of the event, the robability that the event will occur during the next h eriods always has the same value The Beta distribution 8
9 X has a Beta distribution with arameters α and β both of them are ositive) if its df is Γα+β) Γα)Γβ) xα x) β for 0 <x< 0 otherwise The mean and variance of X are EX) = VarX) = α α + β, αβ α + β) α + β +) In articular, if α = β =, then the beta distribution is just the uniform distribution on [0, ] 3 Large-Samle Distribution Theory Our following exosition is based on one-dimensional random variable, but they can be easily extended to the multi-dimensional case We review several concets first: A random samle is a list of indeendently and identically distributed random variables iid) Given a random samle {X,X,,X n } from some distribution with mean μ and variance σ,wecandefine some samle moments: the samle mean is P n X n = X i n the samle variance is P n Sn = X i X n ) n notice that the samle moments are themselves random variables So we can comute their momments: E X) = μ, Var X) = σ n, ESn) = σ 9
10 3 Weak Law of Large Numbers Definition Let {x n } be a sequence of random variables and c be a constant Then x n converges in robability to c or x n c if for any ε>0 lim n Pr x n c >ε)=0 intuitively seaking, it requires that x n concentrates around c as n becomes sufficiently large but it does not require all realizations of x n to be close to c It can tolerate that some realization of x n maybeveryfarawayfromc, but the robability of that event should tend to zero For examle, if x n satisfies Prx n =0)= n and Prx n = n) = n, x n still converges in robability to zero Slutsky Theorem: ifx n Weak Law of Large Numbers: c and g is a continuous function, then gx n ) gc) Let {x n } be a sequence of iid random variables with mean μ and E x n ) < Let x n = n P n x i Then x n μ thistheoremimliesthatifwetakealargesamlefromadistributionforwhich the mean is unknown, then the samle mean will usually be a close estimate of the unknown mean the basic idea of the roof is not difficult Suose the variance of x n is σ Then the Chebyshev inequality imlies that, for any ε>0, Pr x n μ >ε) σ nε since the variance of x n is σ /n It is easy to see that the right-hand side tends to zero as n Definition Let {x n } be a sequence of random variables, and x n has mean μ n and variance σ n Suose μ n c and σ n 0 Thenx n converges in mean square to c Let {x n } n=,, be a sequence of random variables Then But the converse is not true x n converges in mean square to c = x n c 0
11 this result is useful when it is difficult to directly aly the definition of convergence in robability is difficult while we are ready to calculate the mean and variance of a sequence of random variables 3 Central Limit Theorem Lindeberg-Levy) Definition Let {F n } be the sequence of distribution functions of random variables {x n },andletf be the distribution function of x Thenx n converges in distribution to d x or x n x if lim F n = F n at all continuous oints of F for examle, the random variables x n with Prx n =0)= n and Prx n =)= + n converges in distribution to x with Prx =0)=Prx =)= Two useful results similar to the Slutsky theorem: if x n d x and y n c, thenx n y n d cx d d if x n x and g is a continuous function, then gx n ) gx) Central Limit Theorem Lindberg-Levy) Let {x n } be a sequence of iid random variables with finite mean μ and finite variance σ Let x n = n P n x ithen n xn μ) d N 0,σ notice that this result does not assume secific distribution form of x n we have other stronger central limit theorems For examle, under reasonable conditions, we even do not require all random variables have the same mean and variance from WLLN and CLT, we know that the samle mean x n is aroximately anormally distributed random variable with mean μand varianceσ /n The delta method: The delta method is often used to aroximate the distribution of functions of random variables Consider a sequence of random variables {y n } with d nyn μ) N0,σ ) We now want to know the distribution of fy n ) where f is a differentiable function
12 using the Taylor s exansion, we have fy n )=fμ)+f 0 z n )y n μ) for some z n between y n and μ Noticethaty n μ, soz n μ as well) rearrangement yields n [fyn ) fμ)] = nf 0 z n )y n μ) the Slutsky theorem then imlies d n [fyn ) fμ)] N 0, [σf 0 μ)]
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