AMS570 Lecture Notes # Review of Probability (cotiued) Probability distributios. () Biomial distributio Biomial Experimet: ) It cosists of trials ) Each trial results i of possible outcomes, S or F 3) The probability of gettig a certai outcome, say S, remais the same, from trial to trial, say P( S )=p 4) These trials are idepedet, that is the outcomes from the previous trials will ot affect the outcomes of the up-comig trials 5) Let deotes the total # of S amog the trials i a biomial experimet, the ~B(, p), that is, Biomial Distributio with parameters ad p. Its probability desity fuctio (pdf) is f P Here Eg. A exam cosists of 0 multiple choice questios. Each questio has 4 possible choices. Oly is correct. Jeff did ot study for the exam. So he just guesses at the right aswer for each questio (pure guess, ot a educated guess). What is his chace of passig the exam? That is, to make at least 6 correct aswers. Aswer: Yes, this is a biomial experimet with =0, p=0.5, S =choose the right aswer for each questio. Let be the total # of S, the ~B(=0, p=0.5),
( ) ( ) ( ) Beroulli Distributio eroulli(p). It ca take o two possible values, say success (S) or failure (F) with probability p ad (-) respectively. That is P( S ) p P( F ) p Let umber of S, the = { The p f of ca be ritte as f P p ( p) Relatio betwee Beroulli RV ad Biomial RV. () eroulli(p) it is ideed a special case of Biomial radom variable whe (*oly oe trial), that is: ( p) () Let i eroulli(p) i. Furthermore, is are all idepedet. Let i i. The, ( p) (***Exercise, prove this!). Note: *** This liks directly to the Biomial Experimet with i deotes the umber of S for the i trial.
Basics of Statistical Iferece.. Distributios, Mathematical Expectatios, Radom Variables Eg. Let be the height of radomly selected male from etire populatio of adults U.S, the Populatio distributio : ~ N(, ), where is the populatio mea, is the populatio variace, ad N stads for ormal distributio. Its pdf is ( x ) f ( x) e, x (*Note: the Normal distributio is our ext distributio to review!) Let,,, be a radom sample, the,,, are idepedet to each other, ad each follows the same distributio as the populatio distributio That is, the i s are idepedetly, ad idetically distributed (i.i.d.) I. Cumulative distributio fuctio (cdf) If is cotiuous: x F ( x) P( x) f ( t) dt If is discrete: F(x) = P P( t) t II. Probability desity fuctio (pdf) Cotiuous radom variable d f ( x) [ F( x)]' F( x) dx 3
P ( a b) f ( x) dx b a Discrete radom variable (for which the p.d.f. is also called the probability mass fuctio, p.m.f.) f ( x) P( x) P(a b) P a b. Mathematical Expectatio. Special case: Cotiuous radom variable: E[ g( )] g( x) f ( x) dx Discrete radom variable: all E[ g( )] g( x) P( x) ) (populatio) Mea: E( ) x f ( x) dx ) (populatio) Variace: E[( ) ] ( x ) f ( x) dx E( ) [ E( )] *** The above mea ad variace formulas are for cotiuous. Replace the itegral with the summatio (over all possible values of ), if is discrete. Var E E [( ) ] ( ) E E E E ( E ) 3) Momet geeratig fuctio (mgf): 4
Defiitio: Suppose is a cotiuous radom variable (R.V.) with probability desity fuctio (pdf) f(x). The momet geeratig fuctio (mgf) of is defied as M t tx ( t) E( e ) e f ( x) dx, where t is a parameter. (*If is discrete, chage the itegral to summatio.) How to use the mgf to geerate the populatio momets? d First populatio momet: E( ) M ( t) t0 dt Secod populatio momet: d E M t dt t0 I geeral, the k th populatio momet is: properties.) (Exercise, prove the above momet geeratig For ormal distributio, ~ N(, ), ( x ) f ( x) e, x M ( t) e tx f ( x) dx e t t 5
Theorem. If, are idepedet, the f ( x, x ) f ( x ) f ( x ), Theorem. If, are idepedet, the M t M t M t Proof., are idepedet iff (if ad oly if) f ( x, x ) f ( x ) f ( ), thus, x M t E t( ) ( ) ( t x x e ) e f, ( x, x ) dx dx tx tx e f ( x ) dx e f ( x ) dx = M t) M ( t Theorem. Uder regularity coditios, there is a - correspodece betwee the pdf ad the mgf of a give radom variable. That is, pdf f ( x) mgf M ( t). Note: Oe could use this property to idetify the probability distributio based o the momet geeratig fuctio. Special mathematical expectatios for the biomial RV.. Let ~B(, p), please derive the momet geeratig fuctio (m.g.f.) of. Please show the etire derivatio for full credit. m.g.f. of 6
[ ] Theorem. (a b) ( )a b. I additio, we have (*make sure you kow how to derive these expectatios): Exercise: p ar p( p). Please show that the summatio of idepedet Beroulli radom variables each with success probability p, will follow B(, p). Please show the etire derivatio for full credit.. Let, be two idepedet radom variables followig biomial distributios B(, p) ad B(, p) respectively. Please derive the distributio of +. Please show the etire derivatio for full credit Solutio:. Hit: (). the mgf of the Beroulli(p) RV is: (). Let,,, be i.i.d. Beroulli(p), ad let i i 7
The, ( i i) ( ) [ ] (3). Sice the above mgf is the same as the mgf for B(,p), we claim that we have prove this problem Review: You are visitig a radomly selected family with childre (twis ot icluded). You kock o the door ad a boy opes the door (assumig each child has equal chace to ope the door). What is the probability that the other child is a girl? 8
Solutio: Let BG, GG, BB deote the evets that a radomly selected family of two childre has oe boy ad oe girl (BG), two girls (GG) ad two boys (BB), respectively. The we have: 9