Stochastic Processes - lesson 2
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1 Stochastic Processes - lesson 2 Bo Friis Nielsen Institute of Mathematical Modelling Technical University of Denmark 2800 Kgs. Lyngby Denmark bfn@imm.dtu.dk
2 Outline Basic probability theory (from last lesson) Bo Friis Nielsen 8/
3 Outline Basic probability theory (from last lesson) Random variables Bo Friis Nielsen 8/
4 Outline Basic probability theory (from last lesson) Random variables Discrete random variables Bo Friis Nielsen 8/
5 Outline Basic probability theory (from last lesson) Random variables Discrete random variables Continouos random variables Bo Friis Nielsen 8/
6 Outline Basic probability theory (from last lesson) Random variables Discrete random variables Continouos random variables Reading recommendations Bo Friis Nielsen 8/
7 Basic formulas in probability theory Sample value/outcome Event ω A, B Sample space Ω Set of possible events Complementary event Ā = Ω A Union A B The outcome is in at least one of A and B Joint event - intersection A B The outcome is in both A and B Conditional event A B The event A knowing that B has occurred The empty or impossible event See also table 1.1 in Grimmett and Stirzaker page 4 Bo Friis Nielsen 8/
8 P {Ω} = 1 P { } = 0 Bo Friis Nielsen 8/
9 P {Ω} = 1 P { } = 0 P {Ā} = 1 P {A} Bo Friis Nielsen 8/
10 P {Ω} = 1 P { } = 0 P {Ā} = 1 P {A} P {A B} = P {A} + P {B} P {A B} Bo Friis Nielsen 8/
11 P {Ω} = 1 P { } = 0 P {Ā} = 1 P {A} P {A B} = P {A} + P {B} P {A B} Conditional probability Bo Friis Nielsen 8/
12 P {Ω} = 1 P { } = 0 P {Ā} = 1 P {A} P {A B} = P {A} + P {B} P {A B} Conditional probability P {A B} = P {A B} P {B} Bo Friis Nielsen 8/
13 P {Ω} = 1 P { } = 0 P {Ā} = 1 P {A} P {A B} = P {A} + P {B} P {A B} Conditional probability P {A B} = P {A B} P {B} P {A} = P {A B}P {B} Bo Friis Nielsen 8/
14 Law of total probability i B i = Ω Bo Friis Nielsen 8/
15 Law of total probability i B i = Ω B i B j = i j Bo Friis Nielsen 8/
16 Law of total probability i B i = Ω B i B j = i j P {A} = i=1 P {A B i }P {B i } Bo Friis Nielsen 8/
17 Law of total probability i B i = Ω B i B j = i j Independent events P {A} = i=1 P {A B i }P {B i } P {A B} = P {A}P {B} Bo Friis Nielsen 8/
18 Random variables Bo Friis Nielsen 8/
19 Random variables Real valued function of an outcome X(ω) R Bo Friis Nielsen 8/
20 Random variables Real valued function of an outcome X(ω) R Characterised by cumulative distribution function Bo Friis Nielsen 8/
21 Random variables Real valued function of an outcome X(ω) R Characterised by cumulative distribution function F (x) = P {X x} Bo Friis Nielsen 8/
22 Random variables Real valued function of an outcome X(ω) R Characterised by cumulative distribution function F (x) = P {X x} Discrete random variables Bo Friis Nielsen 8/
23 Random variables Real valued function of an outcome X(ω) R Characterised by cumulative distribution function F (x) = P {X x} Discrete random variables Discrete countable sample space Bo Friis Nielsen 8/
24 Random variables Real valued function of an outcome X(ω) R Characterised by cumulative distribution function F (x) = P {X x} Discrete random variables Discrete countable sample space Continouos random variables Bo Friis Nielsen 8/
25 Random variables Real valued function of an outcome X(ω) R Characterised by cumulative distribution function F (x) = P {X x} Discrete random variables Discrete countable sample space Continouos random variables Continouos uncountable sample space Bo Friis Nielsen 8/
26 Discrete random variables Bo Friis Nielsen 8/
27 Discrete random variables Discrete state space Bo Friis Nielsen 8/
28 Discrete random variables Discrete state space Usually integer valued (can always be transformed to be integervalued) Bo Friis Nielsen 8/
29 Discrete random variables Discrete state space Usually integer valued (can always be transformed to be integervalued) Frequency/probability density function (pdf) Bo Friis Nielsen 8/
30 Discrete random variables Discrete state space Usually integer valued (can always be transformed to be integervalued) Frequency/probability density function (pdf) f(x) = P {X = x} Bo Friis Nielsen 8/
31 Discrete random variables Discrete state space Usually integer valued (can always be transformed to be integervalued) Frequency/probability density function (pdf) f(x) = P {X = x} Cumulative distribution function (CDF) Bo Friis Nielsen 8/
32 Discrete random variables Discrete state space Usually integer valued (can always be transformed to be integervalued) Frequency/probability density function (pdf) f(x) = P {X = x} Cumulative distribution function (CDF) F (x) = P {X x} = x t= f(t) Bo Friis Nielsen 8/
33 Example 1 Bernoulli distribution Bo Friis Nielsen 8/
34 Example 1 Bernoulli distribution Example 1 Bernoulli distribution Simplest possible random experiment Bo Friis Nielsen 8/
35 Example 1 Bernoulli distribution Simplest possible random experiment Two possibilites Bo Friis Nielsen 8/
36 Example 1 Bernoulli distribution Simplest possible random experiment Two possibilites Accept/failure Male/female Rain/not rain Bo Friis Nielsen 8/
37 Example 1 Bernoulli distribution Simplest possible random experiment Two possibilites Accept/failure Male/female Rain/not rain One of the possibilities mapped to 1, X(failure) = 0, X(accept) = 1 Bo Friis Nielsen 8/
38 Example 1 Bernoulli distribution Simplest possible random experiment Two possibilites Accept/failure Male/female Rain/not rain One of the possibilities mapped to 1, X(failure) = 0, X(accept) = 1 P {accept} = P {X = 1} = f(1) = p, P {failure} = P {X = 0} = f(0) = 1 p Bo Friis Nielsen 8/
39 Example 1 Bernoulli distribution Simplest possible random experiment Two possibilites Accept/failure Male/female Rain/not rain One of the possibilities mapped to 1, X(failure) = 0, X(accept) = 1 P {accept} = P {X = 1} = f(1) = p, P {failure} = P {X = 0} = f(0) = 1 p X be(p) Bo Friis Nielsen 8/
40 Example 2 binomial distribution Collection of independent Bernoulli experiments Bo Friis Nielsen 8/
41 Example 2 binomial distribution Collection of independent Bernoulli experiments n identical experiments Each with probability p for success Experiments mutually independent Bo Friis Nielsen 8/
42 Example 2 binomial distribution Collection of independent Bernoulli experiments n identical experiments Each with probability p for success Experiments mutually independent A number of practical applications Bo Friis Nielsen 8/
43 Example 2 binomial distribution Collection of independent Bernoulli experiments n identical experiments Each with probability p for success Experiments mutually independent A number of practical applications Number of items passed in quality control Number of male locusts Number of left turning vehicles Bo Friis Nielsen 8/
44 binomial distribution - continued X The total number of successes - sequence less important Bo Friis Nielsen 8/
45 binomial distribution - continued X The total number of successes - sequence less important P {X = x} Bo Friis Nielsen 8/
46 binomial distribution - continued X The total number of successes - sequence less important P {X = x} Sequence FFSSSSFSS has probability (1 p) 2 p 4 (1 p)p 2 Bo Friis Nielsen 8/
47 binomial distribution - continued X The total number of successes - sequence less important P {X = x} Sequence FFSSSSFSS has probability (1 p) 2 p 4 (1 p)p 2 n x sequences with x successes Bo Friis Nielsen 8/
48 binomial distribution - continued X The total number of successes - sequence less important P {X = x} Sequence FFSSSSFSS has probability (1 p) 2 p 4 (1 p)p 2 n x sequences with x successes In conclusion f(x) = P {X = x} = n x p x (1 p) n x Bo Friis Nielsen 8/
49 binomial distribution - continued X The total number of successes - sequence less important P {X = x} Sequence FFSSSSFSS has probability (1 p) 2 p 4 (1 p)p 2 n x sequences with x successes In conclusion f(x) = P {X = x} = n x p x (1 p) n x X B(n, p) Bo Friis Nielsen 8/
50 Binomial distribution n=20 p = 0.5 bincf.lst Bo Friis Nielsen 8/
51 Moments - mean value Bo Friis Nielsen 8/
52 Moments - mean value A measure for the location of the probability mass. Bo Friis Nielsen 8/
53 Moments - mean value A measure for the location of the probability mass. Single descriptor of the distribution. Bo Friis Nielsen 8/
54 Moments - mean value A measure for the location of the probability mass. Single descriptor of the distribution. First order information Bo Friis Nielsen 8/
55 Moments - mean value A measure for the location of the probability mass. Single descriptor of the distribution. First order information The natural value to use if you want to ignore randomness Bo Friis Nielsen 8/
56 Moments - mean value A measure for the location of the probability mass. Single descriptor of the distribution. First order information The natural value to use if you want to ignore randomness Expected values Theoretical average Bo Friis Nielsen 8/
57 Moments - mean value A measure for the location of the probability mass. Single descriptor of the distribution. First order information The natural value to use if you want to ignore randomness Expected values Theoretical average Long term average Bo Friis Nielsen 8/
58 Mathematical definition of mean value E(X) = x= xf(x) Bo Friis Nielsen 8/
59 Mathematical definition of mean value E(X) = x= xf(x) Properties Bo Friis Nielsen 8/
60 Mathematical definition of mean value E(X) = x= xf(x) Properties E(aX + b) = ae(x) + b Bo Friis Nielsen 8/
61 Mathematical definition of mean value E(X) = x= xf(x) Properties E(aX + b) = ae(x) + b E(X + Y ) = E(X) + E(Y ) Bo Friis Nielsen 8/
62 Mathematical definition of mean value E(X) = x= xf(x) Properties E(aX + b) = ae(x) + b E(X + Y ) = E(X) + E(Y ) Example Bernoulli E(X) = 0 (1 p) + 1 p = p Bo Friis Nielsen 8/
63 Example binomial E(X) = np (mean of sum of n identical distributed variables) Bo Friis Nielsen 8/
64 Example binomial E(X) = np (mean of sum of n identical distributed variables) Alternatively directly from definition E(X) = n xf(x) = n x=0 x=0 x n x p x (1 p) n x = Bo Friis Nielsen 8/
65 Example binomial E(X) = np (mean of sum of n identical distributed variables) Alternatively directly from definition E(X) = n xf(x) = n x=0 x=0 x n x p x (1 p) n x = n x=0 x n! x!(n x)! px (1 p) n x = n x=1 n! (x 1)!(n x)! px (1 p) n x Bo Friis Nielsen 8/
66 Example binomial E(X) = np (mean of sum of n identical distributed variables) Alternatively directly from definition E(X) = n xf(x) = n x=0 x=0 x n x p x (1 p) n x = n x=0 x n! x!(n x)! px (1 p) n x = n x=1 n! (x 1)!(n x)! px (1 p) n x n x=1 np (n 1)! (x 1)!((n 1) (x 1))! px 1 (1 p) (n 1) (x 1) Bo Friis Nielsen 8/
67 Example binomial E(X) = np (mean of sum of n identical distributed variables) Alternatively directly from definition E(X) = n xf(x) = n x=0 x=0 x n x p x (1 p) n x = n x=0 x n! x!(n x)! px (1 p) n x = n x=1 n! (x 1)!(n x)! px (1 p) n x n x=1 np (n 1)! (x 1)!((n 1) (x 1))! px 1 (1 p) (n 1) (x 1) np n 1 y=0 (n 1)! y!((n 1) y)! py (1 p) n 1 y = np Bo Friis Nielsen 8/
68 Variance Bo Friis Nielsen 8/
69 Variance Measures irregularity of distribution - variation Bo Friis Nielsen 8/
70 Variance Measures irregularity of distribution - variation Second order information Bo Friis Nielsen 8/
71 Mathematical definition Bo Friis Nielsen 8/
72 Mathematical definition V (X) = x= (x E(X)) 2 f(x) Bo Friis Nielsen 8/
73 Mathematical definition V (X) = x= (x E(X)) 2 f(x) Properties Bo Friis Nielsen 8/
74 Mathematical definition V (X) = x= (x E(X)) 2 f(x) Properties V (ax + b) = a 2 V (X) Bo Friis Nielsen 8/
75 Mathematical definition V (X) = x= (x E(X)) 2 f(x) Properties V (ax + b) = a 2 V (X) V (X + Y ) = V (X) + V (Y ) if X and Y are independent Bo Friis Nielsen 8/
76 Example Bernoulli/Binomial Bo Friis Nielsen 8/
77 Example Bernoulli/Binomial X be(p) Bo Friis Nielsen 8/
78 Example Bernoulli/Binomial X be(p) V (X) = 1 x=0 (x E(X)) 2 f(x) = (0 p) 2 (1 p) + (1 p) 2 p = p(1 p) Bo Friis Nielsen 8/
79 Example Bernoulli/Binomial X be(p) V (X) = 1 x=0 (x E(X)) 2 f(x) = (0 p) 2 (1 p) + (1 p) 2 p = p(1 p) Y B(n, p) Bo Friis Nielsen 8/
80 Example Bernoulli/Binomial X be(p) V (X) = 1 x=0 (x E(X)) 2 f(x) = (0 p) 2 (1 p) + (1 p) 2 p = p(1 p) Y B(n, p) V (Y ) = np(1 p) Bo Friis Nielsen 8/
81 Example 4 Poisson distribution Unlimited number of occurrences Bo Friis Nielsen 8/
82 Example 4 Poisson distribution Unlimited number of occurrences Number of accidents during a year in a traffic crossing Number of weak points in steel plate Number of sick days in large company Number of fish in trawl catch Bo Friis Nielsen 8/
83 Example 4 Poisson distribution Unlimited number of occurrences Number of accidents during a year in a traffic crossing Number of weak points in steel plate Number of sick days in large company Number of fish in trawl catch Limiting distribution for binomial distribution with n and np = µ fixed Bo Friis Nielsen 8/
84 Example 4 Poisson distribution Unlimited number of occurrences Number of accidents during a year in a traffic crossing Number of weak points in steel plate Number of sick days in large company Number of fish in trawl catch Limiting distribution for binomial distribution with n and np = µ fixed f(x) = P {x occurrences} = µx x! e µ Bo Friis Nielsen 8/
85 Example 4 Poisson distribution Unlimited number of occurrences Number of accidents during a year in a traffic crossing Number of weak points in steel plate Number of sick days in large company Number of fish in trawl catch Limiting distribution for binomial distribution with n and np = µ fixed f(x) = P {x occurrences} = µx x! e µ Mean value E(X) = µ, variance V (x) = µ Bo Friis Nielsen 8/
86 Reading recommendations Generally you should read in order to grasp concepts and get an intuitive understanding of the feel. You should be able to understand formulas to a level such that you can apply them in a proper context for Tuesday September 5: and 1.7 for Friday September 8 and Tuesday September 12, read Chapter 2 lightly and Chapter 3 section: You can skip proofs if you like. For Friday September 15, read Bo Friis Nielsen 8/
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