Chapter 2: Elements of Probability Continued

Transcription

1 MSCS6 Chapter : Elements of Probability Continued Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science Copyright 7 by

2 MSCS6 Agenda.7 Chebyshev s Inequality and the law of Large Numbers.8 Some Discrete Random Variables.9 Continuous Random Variables

3 MSCS6.7 Chebyshev s Inequality and the Laws of Large Numbers Markov s Inequality If X takes on only nonnegative values, then for any value a>, P{ X a} EX [ ] a Proof a, if X a Let Y, if X a X Y, because X it follows that and E[ X ] E[ Y ] ap{ X a}. 3

4 MSCS6.7 Chebyshev s Inequality and the Laws of Large Numbers Chebyshev s Inequality If X is an RV having mean μ and variance σ, then for k>, P{ X k} k Proof: Since (X - μ) /σ is a nonnegative RV with mean E ( X ) P k k ( X ) E[( X ) ]., then from Markov s inequality P{ X a} EX [ ] a 4

5 MSCS6.8 Some Discrete Random Variables Binomial A random variable X has a discrete binomial probability mass function, X~binomial(n,p) if n! i ni P{ X i} p ( p) i,..., n i!( n i)! where, p and n,,3,.... n = number of trials or times we repeat the eperiment. i = the number of successes out of n trials. p = the probability of success on an individual trial. 5

6 MSCS6.8 Some Discrete Random Variables Binomial It can be shown that n n! i E[ X ] i p ( p) i!( n i)! i np ni n i ni var[ X ] ( i np) p ( p) i n! i!( n i)! np( p) 6

7 MSCS6 Binomial: n=5; p=/;num=^4; =binornd (n,p,num,); mean() var() hist(,6) True Simulated n! i P( X i) p ( p) i!( n i)! ni Can also find and plot ECDF

8 MSCS6.8 Some Discrete Random Variables Poisson A random variable X has a discrete Poisson probability mass function, X~Poisson(λ) if i P{ X i} e i! where,. λ is called the intensity parameter i,..., n 8

9 MSCS6.8 Some Discrete Random Variables Poisson It can be shown that n i E[ X ] i e i! i n i X i e var[ ] ( ) i i! 9

10 MSCS6 Poisson: lam=5;num=^4; =poissrnd(lam,num,); mean() var() hist(,5) True Simulated P( X i) i e i! Can also find and plot ECDF

11 MSCS6.8 Some Discrete Random Variables Geometric A random variable X has a discrete geometric probability mass function, X~geometric(p) if n,,... P{ X n} p( p) n where p and n,,3,.... X represents the first successful trial in a binomial eperiment with probability of success p.

12 MSCS6.8 Some Discrete Random Variables Geometric It can be shown that E[ X ] np( p) n n / p n var[ X ] ( n / p) p( p) ( p) / n p

13 MSCS6 Geometric: P( X n) p( p) n p=/;num=^4; = geornd(p,num,)+; mean() var() hist(,) True Simulated Can also find and plot ECDF

14 MSCS6.8 Some Discrete Random Variables Negative Binomial A random variable X has a discrete negative binomial probability mass function, X~NB(p,r) if n r nr P{ X n} p ( p) r r,..., n where p and n,,3,.... X represents the number of trials needed to amass a total of r successes from a binomial eperiment with probability of success p. 4

15 MSCS6.8 Some Discrete Random Variables Negative Binomial It can be shown that E[ X ] n r n p ( p) nr r r / p nr n r r( p) / p r nr var[ X ] ( n r / p) p ( p) nr 5

16 MSCS6.9 Continuous Random Variables Uniform A random variable has a continuous uniform probability density function, ~uniform(a,b) if f( ) b a ab, where,,., a [ a, b] b b-a a b 6

17 MSCS6.9 Continuous RVs Uniform f() b-a The CDF of the continuous uniform PDF is a a F( ) [ a, b] b a b F() a b a b 7

18 MSCS6.9 Continuous Random Variables Uniform It can be shown that b d b a b a b a b a ( b a) a d ( b a) 8

19 MSCS6 Uniform: f( ) b a a=;,b=;,num=^4; =a+(b-a)*rand(num,); mean() var() hist(,5) True Simulated f( a,b) Can also find and plot ECDF

20 MSCS6 Uniform: f( ) b a F().5 ecdf() a=;,b=;, y =cdf('unif',(:.:),a,b) plot((:.:),y) [F,]=ecdf(); stairs(,f,'linewidth',)

21 MSCS6.9 Continuous Random Variables Normal A random variable has a continuous normal probability density function, ~normal(μ,σ ) if f (, ) ( ) where, and. EX [ ] e var( X ), f() =,= =,= =,=

22 MSCS6 Normal: The CDF of the continuous normal PDF is ( t ) e F( ) dt ( ) t f() F().5 No closed form analytic solution

23 MSCS6 Normal: =(:.:)'; mu=[,,];, sigma=[,,3]; figure() hold on for count=:length(sigma) y = normcdf(,mu(count),sigma(count)); if count== plot(,y,'r','linewidth',) elseif count== plot(,y,'g','linewidth',) elseif count==3 plot(,y,'b','linewidth',) end end lim([ ]) F() f( ) e ( ) =,= =,= =,=

24 MSCS6 Normal: mu=;,sigma=;,num=^4; =normrnd(mu,sigma,num,); mean() var() hist(,35) True f( ) Simulated e ( ) Can also find and plot ECDF

25 MSCS6 Normal: f( ) ( ) mu=;,sigma=; y=normcdf((:.:5),mu,sigma); plot((:.:5),y, r ) hold on [F,]=ecdf(); stairs(,f,'linewidth',) e ecdf() ecdf()

26 MSCS6.9 Continuous Random Variables Eponential A random variable has a continuous eponential probability density function, ~eponential(λ) if f( ) e, where,. EX [ ] var( X ) 6

27 MSCS6 Eponential: =(:.:5)'; lambda=[.5,.5,,,5];, beta=./lambda; figure() hold on for count=:length(lambda) y = eppdf(,beta(count)); if count== plot(,y,'r','linewidth',) elseif count== plot(,y,'g','linewidth',) elseif count==3 plot(,y,'b','linewidth',) elseif count==4 plot(,y,'m','linewidth',) elseif count==5 plot(,y,'k','linewidth',) end end ylim([ 3]), lim([ 5]) Matlab uses gamma dist def f() f ( ) e =.5 =.5 =. =. =

28 MSCS6 Eponential: f ( ) e.8 The CDF of the continuous eponential PDF is F( ) e Note that there is a closed form solution! f() F()

29 MSCS6 Eponential: =(:.:5)'; lambda=[.5,.5,,,5];, beta=./lambda; figure() hold on for count=:length(lambda) y = epcdf(,beta(count)); if count== plot(,y,'r','linewidth',) elseif count== plot(,y,'g','linewidth',) elseif count==3 plot(,y,'b','linewidth',) elseif count==4 plot(,y,'m','linewidth',) elseif count==5 plot(,y,'k','linewidth',) end end lim([ 5]),ylim([ ]) F() f ( ) e =.5 =.5 = = =

30 MSCS6 Eponential: f ( ) e lambda=;,num=^4; =eprnd(/lambda,num,); mean() var() hist(,35) 5 5 True Simulated Can also find and plot ECDF

31 MSCS6 Eponential: f ( ) e lambda=; y=epcdf((:.:5),/lambda); plot((:.:5),y, 'r') hold on [F,]=ecdf(); stairs(,f,'linewidth',) ecdf() ecdf()

32 MSCS6 Net Time.9 Continuous Random Variables Memoryless Property and Poisson Processes. Conditional Epectation and Variance 3

33 MSCS6 Homework : Chapter : # 7, 3, 9, 3, 36,