Discete Distibutions - Chapte Discete Distibutions Pobability Distibution (Pobability Model) If a balanced coin is tossed, Head and Tail ae equally likely to occu, P(Head) = = / and P(Tail) = = /. Rando Vaiables of Discete Type / Pobability Ba Chat Total pobability is. / Pobability Line Chat Head Tail Head Tail Rando Vaiable Given a ando expeient with an outcoe space S, a function that assigns to each outcoe s in S one eal nube (s) = x, is called a ando vaiable. Usually is denoted by a capital lette., Y, Z,... Vey useful fo atheatically odeling the distibution of vaiables. Discete Rando Vaiable : (Toss a balanced coin) =, if Head occus, and / = 0, if Tail occus. P(Head) = P( = ) = P() = P(Tail) = P( = 0) = P(0) = Pobability ass function: f(x) = P( = x) =, if x = 0,, and 0 elsewhee. Pobability Ba Chat 0 4 Discete Rando Vaiable Why Rando Vaiable? Discete Rando Vaiable: A ando vaiable assues discete values by chance. A siple atheatical notation to descibe an event. e.g.: <, = 0,... Matheatical function can be used to odel the distibution though the use of ando vaiable. e.g.: Binoial, Poisson, Noal, 5 6 DD -
Discete Distibutions - Pobability Mass Function The pobability ass function (p..f.) f(x) of a discete ando vaiable is a function that satisfies the following popeties: a) f(x) > 0 b) Σ x S f(x) = c) P( A) = Σ x A f(x), whee A S. 7 Discete Rando Vaiable : What is pobability of getting a nube less than when oll a balanced die? Pobability ass function: (Unifo Distibution) f (x) = P( = x) = /6, if x =,,, 4, 5, o 6, and 0 elsewhee. P( < ) = P( ) =? Answe: /6 = / 8 Unifo Distibution If ando vaiable has a Discete Unifo Distibution ove fist integes, x =,,,, then has a p..f. f(x) =. Checking Distibution (Unifo Distibution) f (x) = P( = x) = /6, if x =,,, 4, 5, o 6, and 0 elsewhee. Actual Data 9 4 5 6 0 Pobability Mass Function If f(x) is a p..f., find the value of c if c = x, fo x =,,. i= = f () + f () + f () = Popety c c c = + + = c ( + = c = c = + ) =x 00 Is it a pofitable insuance peiu? f (x)..65?.75 0 Distibution: f (-00) =. f (-) =, f () =.65, -00 -.65 =. Pobability line chat Pobability Mass Function, if, if, if x = x = x = 00 DD -
Discete Distibutions - Hypegeoetic Distibution Hypegeoetic Distibution Thee ae 4 ed balls and 5 blue balls in a box. If balls ae selected at ando without eplaceent, what is the pobability that two of the ae ed balls? n( Red) P ( Red) = n( S) 4 5 = 9 6 5 = = 84 5 4 Conside a collections of N = N + N objects, N of the ae of the fist kind and N ae of anothe kind. If a ando saple of n objects is selected fo these N objects without eplaceent. The pobability of having exactly x of the fist kind in the saple is N N x n x P( x) = = =, x n, x N, n x N N n 4 Acceptance Sapling Plan A saple of n = 5 pats is to be selected fo a lot of 5 pats. Soe of the pats ae defective. If all 5 pats ae non-defective then the lot will be accepted. The pobability that none of the pats ae defective (acceptance), if N of the pat in the lot of 5 ae defective, is, N 5 N Opeating Chaacteistic Function 0 5 N OC ( p) = P( = 0) =, p =. 5 5 5 N 5 N 0 5 N OC ( p) = P( = 0) =, p =. 5 5 5 Defective ate Pobability of accepting the lot. (A function of defective ate to exaine if the saple size is easonable.) N P OC(p) 0 0/5=.00.000 /5=.04 0.800 /5=.08 0.6 /5=. 0.496 4 4/5=.6 0.8 Too low Too high Defective ate 5 6 Expected Value Chapte Discete Distibutions Epiical study: Play the gae 000 ties, if head tuns you win $, othewise, you win $0. On aveage, how uch oney do you win pe gae?. Matheatical Expectation Outcoe x Fequency Relative Fequency f (x) Head($) 500 Tail ($0) 500 7 Total 000 8 DD -
Discete Distibutions - Expected Value Expected Value Epiical study: Play the gae 000 ties, if head tuns you win $, othewise, you win $0. On aveage, how uch oney do you win pe gae? Aveage = ( 500 + 0 500) / 000 f (x) = 500 / 000 = o = 500/000 + 0 500/000 = + 0 = Epiical study: Play the gae 000 ties, if head tuns you win $, othewise, you win $0. On aveage, how uch oney do you win pe gae? Outcoe, x Head($) Tail ($0) Relative Fequency, f(x) x 9 0 Total.0 Poduct, x f (x) 0 Σ x f (x) Matheatical Expectation If f (x) is the p..f. of a discete ando vaiable, with space S, the atheatical expectation of is E[] = Σ S x f(x) and the atheatical expectation of u() is E[u()] = Σ S u(x) f(x) What is the pobability distibution of olling a die? Pobability ass function: (Unifo Distibution) f (x) = P( = x) = /6, if x =,,, 4, 5, o 6, x f(x) and 0 elsewhee. /6 E[] =? 4 5 6 /6 /6 /6 /6 /6 E[ ] = + + 6 6 6 + 4 + 5 + 6 = = 6 6 6 6 Is it a pofitable insuance peiu? x f(x) x f(x) -00. -00. - -.75 Pobability line chat.65.65 0 Let ando vaiable have the p..f. f (x) = 0, x S, whee x =, 0,,, find E[()]. -00 - The ean of the distibution is E[] = ( 00).+ ( ) +.65 =. (Weighted by pobabilities.) E[] = f( ) + 0 f(0) + f() + f() = + 0 + + = 0 4 DD - 4
Discete Distibutions - E[] = 0 Theoe of Expected Value Let ando vaiable have the p..f. f (x) = 0, x S, whee x =, 0,,. Let u(x) = x, find E[u()]. E[u()]= u( ) f( ) + u(0) f(0) + u() f() + u() f() =( ) f( ) + 0 f(0) + f() + f() = + 0 + + 6 = 5 Matheatics expectation, if exists, satisfies the following popeties: a) If c is a constant, E[c] = c. b)if c is a constant and u is a function, E[c u()] = c E[u()]. c) If c and c ae constants and u and u ae functions, then E[c u () +c u ()] = c E[u ()] +c E[ u ()]. 6 Let ando vaiable have the p..f. f (x) = 0, x S, whee x =, 0,,, find E[ ]. Let ando vaiable have the p..f. f (x) = 0, x S, whee x =, 0,,, find E[ ( )]. E[ ] = ( ) f( ) + (0) f(0) + () f() + () f() E[( )] = E[ ] = f( ) + 0 f(0) + f() + 4 f() = E[ ] E[ ] = + 0 + + 4 = E[ ] E[ ] = = 0 = 7 8 Let have a hypegeoetic distibution is which n objects ae selected fo N = N + N then N N x n x E[ ] x = x S N n N = n N 9 Chapte Discete Distibutions. The Mean, Vaiance and Standad Deviation 0 DD - 5
Discete Distibutions - Mean and Vaiance of a Rando Vaiable If f (x) is the p..f. of a discete ando vaiable, with space S, then the ean of the ando vaiable is μ = E[] = Σ S x f(x) and the vaiance of the ando vaiable is σ = E[( μ) ] = Σ S (x μ) f(x) Standad Deviation = σ =? E[( μ) ] Vaiance of a Rando Vaiable σ = E[ ] μ Poof: σ = E[( μ) ] = E[ μ + μ ] = E[ ] με[] + μ = E[ ] μ + μ = E[ ] μ Let ando vaiable have the p..f. f (x) = 0, x S, whee x =, 0,,, find the vaiance of E[] = 0 E[ ] = σ = E[ ] μ = 0 = 0 = Unifo Distibution Let ando vaiable have a Discete Unifo Distibution ove fist integes, x =,,,, that has a p..f. f(x) =. [ ] = ( ) = = E x f x x x x= x= x= ( + ) + = = ( + )( + ) E[ ] = x = x = x= x= 6 σ = E[( μ) ] = E[ ] μ = 4 Hypegeoetic Distibution Conside a collections of N = N + N objects, N of the ae of the fist kind and N ae of anothe kind. A ando saple of n objects is selected fo these N objects without eplaceent. The ando vaiable is the nube of fist kind in the saple has the ean, μ, and vaiance, σ, N μ = n = np N N N N n N n σ = n = np( p) N N N N 5 Let be a.v. that has a unifo distibution, f (x) = P( = x) = /6, if x =,,, 4, 5, o 6, and 0 elsewhee. Find ean and vaiance of. 6 + Mean = μ = =. 5 Vaiance = σ 6 5 = = 6 DD - 6
Discete Distibutions - Linea Function of a Rando Vaiable Let be a ando vaiable with ean μ and vaiance σ, and let ando vaiable Y be a linea function of the, and Y = a + b, then μ Y = E[ Y ] = E[ a + b] = a E[ ] + b = aμ + b σ = E[( Y μ ) ] = E[( a + b aμ b) ] Y = E[ a Y ( μ ) ] = a σ 7 Let ando vaiable have the p..f. f (x) = 0, whee x =, 0,,, find the ean and vaiance of Y = + μ = 0, σ = fo ealie exaple μ Y = x 0 + = σ Y = x = What is the standad deviation? 8 -th Moent Let be a positive intege, the -th oent about the oigin, if exists, is E [ ] = x x S Let be a positive intege, the -th oent about b, if exists, is Factoial Moent Let be a positive intege, the -th factoial oent, if exists, is E[( ) ] = E[ ( )( )...( + )] E [( b) ] = ( x b) x S 9 40 DD - 7