STAT/MATH 395 PROBABILITY II

Transcription

1 STAT/MATH 395 PROBABILITY II Chapter 6 : Moment Functions Néhémy Lim 1 1 Department of Statistics, University of Washington, USA Winter Quarter 2016

2 of Common Distributions Outline of Common Distributions

3 of Common Distributions Outline of Common Distributions

4 of Common Distributions Definition Let X be a rrv on probability space (Ω, A, P). For a given r N, E[X r ], if it exists, is called the r-th moment of X. In particular, If X is a discrete rrv with pmf p X and X(Ω) is the set of all values that X can take on, then E[X r ] = x r p X (x) (1) x X(Ω) If X is a continuous rrv with pdf f X, then E[X r ] = x r f X (x) dx (2)

5 of Common Distributions Definition Let X be a rrv on probability space (Ω, A, P). For a given r N, E[(X E[X]) r ], if it exists, is called the r-th moment about the mean or r-th central moment of X. In particular, If X is a discrete rrv with pmf p X and X(Ω) is the set of all values that X can take on, then E [(X E[X]) r ] = (x E[X]) r p X (x) (3) x X(Ω) If X is a continuous rrv with pdf f X, then E [(X E[X]) r ] = (x E[X]) r f X (x) dx (4)

6 of Common Distributions Outline of Common Distributions

7 of Common Distributions Definition Let X be a rrv on probability space (Ω, A, P). For a given t R, the moment generating function (m.g.f.) of X, denoted M X (t), is defined as follows M X (t) = E [ e tx ] (5) where there is a positive number h such that the above summation exists for h < t < h. In particular, If X is a discrete rrv with pmf p X and X(Ω) is the set of all values that X can take on, then M X (t) = e tx p X (x) (6) x X(Ω) If X is a continuous rrv with pdf f X, then M X (t) = e tx f X (x) dx (7)

8 of Common Distributions Theorem A moment generating function completely determines the distribution of a real-valued random variable.

9 of Common Distributions Proposition Let X be a rrv on probability space (Ω, A, P). The r-th moment of X can be found by evaluating the r-th derivative of the m.g.f. of X at t = 0. M (r) X (0) = E[X r ] (8) Lemma Let X be a rrv on probability space (Ω, A, P). 1 The expected value of X, if it exists, can be found by evaluating the first derivative of the moment generating function at t = 0 : E[X] = M X (0) (9) 2 The variance of X, if it exists, can be found by evaluating the first and second derivatives of the moment generating function at t = 0 : Var(X) = M X (0) (M X (0)) 2 (10)

10 Outline of Common Distributions of Common Distributions

11 of Common Distributions Assume an instructor recklessly assigns a random grade (integer between 1 and 6) to his students. Let X be the grade of a student of his class.

12 of Common Distributions Assume Federer beats Berdych 70% of the time. Let X be the number of Federer wins the next 5 times they meet.

13 of Common Distributions Definition (Bernoulli process) A Bernoulli or binomial process has the following features : 1 We repeat n N identical trials 2 A trial can result in only two possible outcomes, that is, a certain event E, called success, occurs with probability p, thus event E c, called failure, occurs with probability 1 p 3 The probability of success p remains constant trial after trial. In this case, the process is said to be stationary. 4 The n trials are mutually independent.

14 of Common Distributions Assume a tennis player serves 4 double faults on average. Let X be the number of double faults in the next game he plays.

15 of Common Distributions In France, the galette des rois (King cake) contains a figurine, the fève, hidden in the cake and the person who finds the trinket in his or her slice becomes king/queen for the day. Assume that galettes are cut into 6 identical slices. Let X be the number of galettes you eat before you find the fève.

16 of Common Distributions Distribution Probability Mass Function M X (t) E[X] Var(X) p X (x) = P(X = x) 1 e t e (n+1)t n + 1 n 2 1 Uniform U n n n(1 e t ) 2 12 n N if t 0 ( n ) ( ) Binomial Bin(n, p) p x (1 p) n x 0 pe t n + 1 p np np(1 p) x n N, p (0, 1) for x = 0,..., n λ λx Poisson P(λ) e exp λ(e t 1) λ λ x! λ > 0 for x N Geometric G(p) (1 p) x 1 p pe t 1 1 (1 p)e t p p (0, 1) for x N for t < ln(1 p) 1 p p 2

17 of Common Distributions Distribution Probability Density Function M X (t) E[X] Var(X) f X (x) Uniform U(a, b) 1 b a 1 e tb e ta a + b (b a) 2 [a,b](x) t(b a) 2 12 a, b R with a < b if t 0 { ( ) 2 } } Normal N (µ, σ 2 1 ) σ exp 1 x µ exp {µt + σ2 t 2 µ 2π 2 σ 2 σ 2 µ R, σ > 0 for x R Exponential E(λ) λe λx λ R+ (x) λ t λ λ 2 λ > 0 for t < λ