Probability review (week 2) Solutions

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1 Probability review (week 2) Solutios A. Biomial distributio. BERNOULLI, BINOMIAL, POISSON AND NORMAL DISTRIBUTIONS. X is a biomial RV with parameters,p. Let u i be a Beroulli RV with probability of success p=. The epected value of u i is E(u i )=(p)+(-p)=p (the summatio of each possible outcome weighted by its probability). The epected value of X is the sum of the epected values of each idividual trial (represeted by the Beroulli RV u i ), so it ca be derived by: X = where u i s are Beroulli r.v. s Now for the variace: i= E(X ) = E( = u i u i ) = i= p = p. i= E(u i ) i= σ 2 X = E[(X E(X )) 2 ] = E(X 2 ) (EX ) 2 = E( = [due to the liearity of Epectatio] u i i= j= u j 2 p 2 ) Eu i u j 2 p 2 = p + ( )p 2 2 p 2 = p( p). i,j= I calculatio of i,j= Eu iu j, ote that Eu i u j = Eu 2 i = p if i = j, which there are of such terms i the summatio, ad is equal to E(u i )E(u j ) = p.p = p 2, where there are ( ) of them i the summatio. The MatLab code follows: % This fuctio returs a vector the same size as u where etries % are the biomial with parameters ad p, calculated % at each elemet of u. This tries to mimic Matlab s % built-i fuctio f=( bio,u,,p) fuctio f=my_biomial_(u,,p) % Note the followig from help o the fuctio "factorial": % Sice double precisio umbers oly have about % 5 digits, the aswer is oly accurate for N <= 2. % For how Matlab itself calculates the biomial, see bio.m f=zeros(size(u)); %iitializatio of f for i=:legth(u) if (u(i)>=)&&(u(i)<=) % sice support of biomial(,p) is,,.., f(i)=pˆu(i)*(-p)ˆ(-u(i))*factorial()/(factorial(-u(i))*factorial(u(i))); % usig the relatio for of biomial with parameters,p

2 Ad ow the mai m file, where we fi E[X ] = 5 ad = 6,, 2, 5. Notice the use of stem for illustratig pmf ad stairs for cmf i discrete radom variables. clear all; close all; clc; _vector=[6,,2,5]; i=; for =_vector p=5/; figure() subplot(2,2,i); % stem(:,( bio,:,,p),. ); % cheat lie! stem(:,my_biomial_(:,,p),. ); title([ =,um2str()]); label( ); ylabel( ); grid o; ais([,5,,.5]); figure(2) subplot(2,2,i); % stem(:,( bio,:,,p),. ); % cheat lie! % stem(:,my_biomial_(:,,p),. ); % see below for my_biomial_ code stairs(:,cumsum(my_biomial_(:,,p)), LieWidth,); % usig cumsum title([ =,um2str()]); label( ); ylabel( ); grid o; ais([,5,,]); i=i+; I case you prefer a separate fuctio for biomial : % fuctio takes vector u, ad scalars ad p, % ad returs a vector of the same size as u, where etries % are the biomial with parameters ad p, calculated % at the poits of elemets of u. tryig to mimic Matlab s % built-i fuctio f=( bio,u,,p) % This fuctio calls my_biomial_ ad calculates a cumulative sum % over possible values up to u(i) for each etry of u. fuctio F=my_biomial_(u,,p) F=zeros(size(u)); %iitializatio of F for i=:legth(u) F(i)=sum(my_biomial_(:u(i),,p)); The correspodig ad plots ca be foud i figures ad 2. B. Biomial ad Poisso distributios. For Poisso RV X p with parameter λ, the epected value is calculated below. E(X P ) = k e λ λ k = λe λ kλ k = λe λ d k! k! dλ ( λ k k! ) = d λe λ dλ (eλ ) == λe λ e λ = λ k= k= The Matlab code to plot the Poisso distributio follows. clear all; close all; clc; k=

3 figure lambda=5; =:5; % stem(,( poiss,,lambda),. ); % cheat lie! my_poisso_=ep(-lambda)*(lambda.ˆ)./factorial(); % Note the use of "dot" for elemet-wise operatio % Thus, my_poisso_ is ow a vector with the same size as. stem(,my_poisso_,. ); title([ Poisso Distributio with \lambda =,um2str(lambda)]); label( ); ylabel( ); grid o; ais([,5,,.5]); Graph ca be foud i figure 3 Now we eed to calculate the MSE. A simple trial ad error, such as the followig code reveals that we oly eed to cosider the first values for calculatio of MSE (Note that Poisso has a decliig tail, which mea that as, P(X P = ). ) clear all; close all; clc; =:2; lambda=5; my posso=ep(-lambda)*lambda.ˆ./factorial(); test=my posso<.5; test I fact, more precisely, we should cosider oly the terms 3 : 9, as show i the below code, which also geerates figure 4. close all; clear all; clc; lambda=5; =3:9; _ide=; _vector=[6 2 5]; my_mse=zeros(,4); for =_vector _ide=_ide+; my_poisso_=ep(-lambda)*(lambda.ˆ)./factorial(); my_mse(,_ide)=sum((my_biomial_(,,lambda/)... -my_poisso_).ˆ2.*my_poisso_); stem(_vector,my_mse, * ); label( ); ylabel( MSE ); grid o; ais([,5,,.2]); The results reported i a table I, below. TABLE I MEAN-SQUARE-ERROR BETWEEN PDF OF A POISSON WITH λ = 5 AND BINOMIALS OF (, λ/) FOR = 6,, 2, 5. THE PROBABILITIES IN THE POISSON LESS THAN.5 ARE NEGLECTED. MSE The table above shows the MSE betwee the biomial ad Poisso distributios for = 6,, 2, 5. As we ca see, as icreases, the MSE falls rapidly to zero, idicatig the distributios become more ad more similar for

4 larger s. C. Biomial ad Poisso distributios agai. The pmf of biomial RV X (, λ/p) coverges to the pmf of Poisso RV X p as as show below. p X () = λ lim! ( )(! ( )!! p ( p)! = ( )!! ( λ ) ( λ ) ( )( 2)... ( + ) ( λ = λ )! ( λ ) = λ! ( )( )... ( + )... ( + ) ( λ ) ( λ λ = )! ) ( λ ) ( λ ) ( λ (... ) lim ) ( λ ) By defiitio: Thus, = λ! lim ( λ ) ( λ ) = e λ lim p X () = λ! e λ = p Xp () Please refer to pp of the slides titled block probability review for more iformatio. D. Biomial ad ormal distributios. If Z is a RV with stadard ormal distributio, i.e. N(, ), the accordig to i= Z := i µ σ, X = i= u i has a Normal distributio with mea p ad variace σ, i.e. N(p, σ ). (Note: σ is equivalet to p( p)). The code follows: close all; clear all; clc; _vector=[ 2 5]; p=.5; _ide=; for =_vector _ide=_ide+; mea_ormal=*p; variace_ormal=sqrt(*p*(-p)); =:; %defiig the mea %defiig the variace subplot(3,,_ide); %plottig graph stairs(,[(my_biomial_(,,p)), orm(,mea_ormal,variace_ormal) ]..., LieWidth,2); title([ =,um2str()]); label( ); ylabel( ); leg( Biomial, Normal ); grid o; ais([,5,,]); The resultig graph ca be foud i fig 5

5 E. Normal ad Poisso approimatios. I provide you with a hit for this part: Poisso limit theorem is about accumulatio of icreasigly improbable evets. I particular, ote that for covergece of the distributio of sum of iid Beroulli RV s (which is a biomial RV) to a Poiso distributio with mea (=rate) λ, we eeded the probability of i the Beroulli RV to be p = λ/, ad as, this probability goes to zero. O the other had, i CLT, p is fied. Hece, CLT ad Poisso limit theorem are addressig basically differet limits. I leave more cotemplatio o this matter to you!

6 .5 =6.5 = = = Fig.. biomial pmf for = 6,, 2, 5 ad p = 5/. (part A)

7 =6 = = = Fig. 2. biomial cmf for = 6,, 2, 5 ad p = 5/. (part A).5 Poisso Distributio with λ = Fig. 3. Depictig the pmf of Poisso distributio with λ = 5. Note that the support of a Poisso radom variable is the whole set of {,, 2,...} up to, however, the pmf for oly the first 5 poits is show. Note the similarity with figures for large.

8 MSE Fig. 4. Mea-Square-Error betwee of a Poisso with λ = 5 ad Biomials of (, λ/) for = 6,, 2, 5. The probabilities i the Poisso less tha.5 are eglected. As we see, by icreasig, the MSE rapidly vaishes. (part B) = Biomial Normal =2 Biomial Normal =5 Biomial Normal Fig. 5. of the Biomial r.v. Y for differet s: =, 2, 3, ad its approimatio by a ormal. As we ca see, by icreasig, the ormal distributio is providig a improved approimatio of the Biomial distributio, a fact which follows from Cetral Limit Theorem. (part D)