제3장 주요내용과 질문

제3장 주요내용과 질문

3.2 Discrete Distributions

discrete uniform distribution: its expectation and variance
hypergeometric distribution: [Q] Check that its range is max(0, K-N+M)≤x≤min(K,M). Compare the mean and the variance of hypergeometric distribution with the mean and the variance of binomial distribution.
binomial distribution: Bernoulli trials. [Q] What's the difference between the hypergeometric and the binomial?
Poisson distribution: E(X)=Var(X). Derivation of its mgf using the Taylor series of exp(y). The Poisson approximation to the binomial.
negative binomial distribution: two definitions and their equivalence. [Q] Why the name 'negative binomial'? The variance is a quadratic function of the mean. inverse binomial sampling in Example 3.2.6.
geometric distribution: It is a special case of the negative binomial distribution. 'memoryless' property (cf. exponential distribution).

3.3 Continuous Distributions

uniform distribution: its expectation and variance.
gamma distribution: definition of gamma function Γ(α), Γ(α+1)= αΓ(α), Γ(1/2)=√π. its expectation, variance and mgf.
chi-squared distribution with p degrees of freedom: it is a special case of the gamma distribution, Γ(p/2, 2).
exponential distribution with mean β: it is a special case of the gamma distribution, Γ(1, β), and also a special case of the Weibull distribution. 'memoryless' property (cf. geometric distribution).
Weibull distribution: relationship between the exponential and the Weibull;[Q] How to generate a Weibull random number using an exponential random number? The Weibull distribution is used to model lifetime data. hazard functions (Exercises 3.25 and 3.26)
normal distribution: standard normal.[Q] Why does the normal distribution play a central role in statistics? the proof of ∫ exp(-z²/2) I(0<z<∞) dz = √(π/2) using polar coordinates. normal approximation to the binomial as a special case of the Central Limit Theorem. continuity correction in the normal approximation to the binomial (Example 3.3.2).
beta distribution: beta function B(α,β). relationship between the beta function and the gamma function; B(α,β)=Γ(α)Γ(β)/Γ(α+β). The beta distribution is used to model proportions. shapes (Figures 3.3.3, 3.3.4. monotone, U-shaped, unimodal, symmetric). The uniform is a special member of the beta family. relationship between the beta and the F (Theorem 5.3.8 c). relationship between the gamma and the beta (Exercise 4.24)
Cauchy distribution: It has a thick tail, so that no moments exist. The ratio of two standard normals has a Cauchy distribution (Example 4.3.6).
lognormal distribution: definition. calculation of moments using the mgf of normal. The lognormal distribution is very popular in modeling applications, when the variable of interest is skewed to the right. Its appearance is similar to the gamma (Fig. 3.3.6).
double exponential distribution: symmetric with fat tails, but still retains all of its moments. not differentiable at x=μ.

3.4 Exponential Families

definition of a k-parameter exponential family
[Q] Why do we need exponential family?
The set of x values for which f(x|θ)>0 cannot depend on θ in an exponential family.
re-parameterization and natural parameter: Although natural parameters provide a convenient mathematical formulation, they sometimes lack simple interpretations like the mean and variance. (cf. Example 3.4.6)
[Q] Why do we re-parameterize exponential family to have the natural parameter space? - cf. Exercise 3.32
full (or regular) exponential family vs. curved exponential family

3.5 Location and Scale Families

location family and location parameter: If X is a random variable with pdf f(x-μ), then X may be represented as X=Z+μ, where Z is a random variable with pdf f(z) (cf. Theorem 3.5.6). two situations where a location family might be an appropriate model. 'threshold parameter'.
scale family
location-scale family. Theorem 3.5.6.

3.6 Inequlities and Identities

This section will be treated in Section 4.7

Miscellanea

the Poisson postulates

제3장 연습문제(숙제)

4 5 7 11 13 17 20 22 23 24 25 26 28 32 38 39 40

<24.e번 문제 힌트>

Gumbel 확률변수 Y의 기대값과 분산을 구하는 문제인데, Y=α - γlogX 이므로 W=-logX의 평균과 분산을 구해서 E(Y)=α + γE(W), Var(Y)=γ²Var(W)임을 이용하면 계산이 덜 복잡해진다. W의 평균과 분산은 W의 mgf를 이용하여 구한다. 이 때 digamma function과 polygamma function이 필요한데, digamma function 은

ψ(z)=d logΓ(z)/dz = Γ'(z)/Γ(z), z>0

로 정의되고, polygamma function은

ψ⁽ⁿ⁾(z)=d ⁿψ(z)/dzⁿ , n=1,2,...

로 각각 정의된다. 이 때,

ψ(1)=-γ₀=-0.577215... (γ₀를 Euler 상수라 부름)

ψ⁽¹⁾(1)=ψ'(1)=π²/6

임이 알려져 있는데, 이를 이용하면 W의 mgf로부터 W의 기대값과 분산을 구할 수 있다