1 Random Variable. Why Random Variable? Discrete Random Variable. Discrete Random Variable. Discrete Distributions - 1 DD1-1
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1 Rando Vaiable Pobability Distibutions and Pobability Densities Definition: If S is a saple space with a pobability easue and is a eal-valued function defined ove the eleents of S, then is called a ando vaiable. [ (s) = x ] Rando vaiables ae usually denoted by capital lette., Y, Z,... Lowe case of lettes ae usually fo denoting a eleent o a value of the ando vaiable. Discete Rando Vaiable Why Rando Vaiable? : (Toss a balanced coin) =, if Head occus, and = 0, if Tail occus. P(Head) = P( = ) = P() =.5 P(Tail) = P( = 0) = P(0) =.5 / Pobability ass function: f(x) = P( = x) =.5, if x = 0,, and 0 elsewhee. Pobability Ba Chat 0 3 A siple atheatical notation to descibe an event. e.g.: < 3, = 0,... Matheatical function can be used to odel the distibution though the use of ando vaiable. e.g.: Binoial, Poisson, Noal, 4 Discete Rando Vaiable Discete Rando Vaiable: A ando vaiable assues discete values by chance. Pobability Mass Function (o Pobability Distibution) 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) S xs f(x) = c) P( A) = S xa f(x), whee A S. 5 6 DD -
2 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 =,, 3, 4, 5, o 6, and 0 elsewhee. Actual Data Pobability Mass Function If f(x) is a p..f., find the value of c if c x, fo x,, i f () f () f (3) Popety c c c 3 3 c ( ) c c 9 =x Is it a pofitable insuance peiu? f (x) ? Distibution: f (-00) =. f (-) =.5, f () =.65, Pobability line chat Pobability Mass Function, if, if, if x x - x Hypegeoetic Distibution Hypegeoetic Distibution Thee ae 4 ed balls and 5 blue balls in a box. If 3 balls ae selected at ando without eplaceent, what is the pobability that two of the ae ed balls? n( Red) P( Red) n( S) 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 DD -
3 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 N 5 - N 0 5 N OC( p) P( 0), p 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= /5= /5= /5= /5= Too low Too high Defective ate 3 4 Expected Value Discete Distibutions Epiical study: Play the gae 000 ties, if head tuns you win $, othewise, you win $0. On aveage, how uch oney do you win pe gae? Matheatical Expectation Outcoe x Fequency Relative Fequency f (x) Head($) Tail ($0) Total 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 = ( ) / 000 = 500 / 000 =.5 o = 500/ /000 = =.5 x f (x) 7 Expected Value Epiical study: Play the gae 000 ties, if head tuns you win $, othewise, you win $0. On aveage, how uch oney do you win pe gae? Outcoe, x Relative Fequency, f(x) Poduct, x f (x) Head($).5.5 Tail ($0) Total.0.5 S x f (x) 8 DD - 3
4 Matheatical Expectation If f (x) is the p..f. of a discete ando vaiable, with space S, the atheatical expectation of is ] = S S x f(x) and the atheatical expectation of u() is u()] = S S u(x) f(x) 9 What is the pobability distibution of olling a die? Pobability ass function: (Unifo Distibution) f (x) = P( = x) = /6, if x =,, 3, 4, 5, o 6, x f(x) and 0 elsewhee. /6 ] =? /6 3 /6 4 /6 5 /6 6 /6 ] Is it a pofitable insuance peiu? x f(x) x f(x) Pobability line chat Let ando vaiable have the p..f. f (x) = 0.5, x S, whee x = -, 0,,, find ()] The ean of the distibution is ] = (-00).+ (-) = -3. (Weighted by pobabilities.) ] = - f(-) + 0 f(0) + f() + f() = = 0.5 ] = 0.5 Theoe of Expected Value Let ando vaiable have the p..f. f (x) = 0.5, x S, whee x = -, 0,,. Let u(x) = 3x, find u()]. u()]= u(-) f(-) + u(0) f(0) + u() f() + u() f() =3(-) f(-) f(0) + 3 f() + 3 f() = =.5 3 Matheatics expectation, if exists, satisfies the following popeties: a) If c is a constant, c] = c. b)if c is a constant and u is a function, c u()] = c u()]. c) If c and c ae constants and u and u ae functions, then c u () +c u ()] = c u ()] +c u ()]. 4 DD - 4
5 Let ando vaiable have the p..f. f (x) = 0.5, x S, whee x = -, 0,,, find ]. Let ando vaiable have the p..f. f (x) = 0.5, x S, whee x = -, 0,,, find ( )]. ] = (-) f(-) + (0) f(0) + () f() + () f() ( - )] = - ] = f(-) + 0 f(0) + f() + 4 f() = ] - ] = = ] - ] =.5 = = Let have a hypegeoetic distibution is which n objects ae selected fo N = N + N then N N x n x ] x - xs N n N n N 7 Discete Distibutions 3 The Mean, Vaiance and Standad Deviation 8 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 = ] = S S x f(x) and the vaiance of the ando vaiable is s = ( - ) ] = S S (x - ) f (x) Standad Deviation = s =? ( - ) ] 9 Vaiance of a Rando Vaiable s = ] - Poof: s = ( - ) ] = - + ] = ] - ] + = ] - + = ] - 30 DD - 5
6 Let ando vaiable have the p..f. f (x) = 0.5, x S, whee x = -, 0,,, find the vaiance of ] = 0.5 ] =.5 s = ] - = = =.5 3 Unifo Distibution Let ando vaiable have a Discete Unifo Distibution ove fist integes, x =,,,, that has a p..f. f(x) =. ] x x x x x x ( ) ( )( ) ] x x x x 6 - s ( - ) ] ]- 3 Hypegeoetic Distibution Let be a.v. that has a unifo distibution, f (x) = P( = x) = /6, if x =,, 3, 4, 5, o 6, and 0 elsewhee. Find ean and vaiance of. 6 Mean = = 3. 5 Vaiance = s = 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,, and vaiance, s, N n np N N N N - n N - n s n np( - p) N N N - N - 34 Linea Function of a Rando Vaiable Let be a ando vaiable with ean and vaiance s, and let ando vaiable Y be a linea function of the, and Y = a + b, then Y] a b] a ] b a b Y s ( Y - ) ] ( a b - a -b) ] Y a Y ( - ) ] a s 35 Let ando vaiable have the p..f. f (x) = 0.5, whee x = -, 0,,, find the ean and vaiance of Y = 3 + = 0.5, s =.5 fo ealie exaple Y = 3 x = 3.5 s Y = 3 x.5 =.5 What is the standad deviation? 36 DD - 6
7 -th Moent Factoial Moent Let be a positive intege, the -th oent about the oigin, if exists, is E [ ] x xs Let be a positive intege, the -th oent about b, if exists, is E [( -b) ] ( x -b) xs 37 Let be a positive intege, the -th factoial oent, if exists, is ( ) ] ( -)( - )...( - )] 38 Let be a discete ando vaiable (.v.) that has a p..f. f(x) = x/6, fo x =,, 3; f(x) = 0 elsewhee. Find ] = ] = s = + ] = Va[ + ] = 39 DD - 7
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