5 Probability densities
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1 5 Probbility densities 5. Continuous rndom vribles 5. The norml distribution 5.3 The norml pproimtion to the binomil distribution 5.5 The uniorm distribution 5. Joint distribution discrete nd continuous Introduction 3 Probbility 4 Probbility distributions 5 Probbility densities Orgniztion & description 6 Smpling distributions 7 Inerences.. men 8 Compring tretments 9 Inerences.. vrinces A Rndom processes
2 5.5 The Uniorm Distribution or Uniorm Distribution elsewhere d d F ' d 3 ' 4 3
3 5. Joint Distributions Discrete & Continuous 3 We hve considered cses in which there is single outcome in ech tril. Hence, we use single rndom vrible, sy X, nd represent the sttisticl chrcteristics o X by its probbility density unction or pd p.d... In some cses, there re two outcomes in ech tril. These cses require two rndom vribles, sy X nd X, nd we need to represent the sttisticl chrcteristics o X nd X, by their joint probbility density unction,.
4 Joint p.d.. o continuous rndom vribles 4
5 Joint Joint p.d.. p.d.. o continuous rndom vribles o continuous rndom vribles 5 De. The joint probbility density unction o rndom vribles X nd X is unction, tht possesses the properties,, d d ii i,,, b b d d b b P iii X X DTh i l b bili d it ti d i bl X De. The mrginl probbiliy density unctions o rndom vribles X nd X re respectively.,, d d &
6 6 De. The mrginl probbiliy density unctions o rndom vribles X nd X re, d &, d.,
7 e.g. Joint p.d.. o continuous rndom vribles 7 The joint p.d.. o rndom vribles X & X is, = Find or the joint p.d.. b Find P X, X 3 nd c Find P X, X Sln:, d d e d 3 e d 3 e 3 6 note: e, othewise b P X, X 3 6 e d e d e e e e 3 =.3 d e 3 6 e 6 d d 4 e c PX, X e =.5
8 Joint p.d.. o continuous rndom vribles 8 De. The mrginl probbiliy density unctions o rndom vribles X nd X re, d &, d. e.g. The joint p.d.. o rndom vribles X & X is, = Find the mrginl probbility density unction o X. Sln: Integrting out, we get 3, 6e othewise 3 6e d or othewise note: e d e e othewise
9 De. The joint cumultive distribution unction is e.g. F, u, v du dv P X, X F, so,. The joint p.d.. o rndom vribles X & X is, = 3, 6e othewise 9 Find the joint cumultive distribution unction F, b Find P X, X. Sln: F u3v 6e du dv or,, 3 otherwise e e or, otherwise note: e d e b PX, X = F, e e. 86 3
10 De. The conditionl probbiliy density o rndom vrible X being given, provided tht rndom vrible X being is. e.g. The joint p.d.. o rndom vribles X & X is, = 3, othewise Find the conditionl probbiliy density o rndom vrible X being given tht rndom vrible X tes on the vlue. Sln: The mrginl p.d.. o X is 4 d 3 3 or & otherwise conditionl 4 or & 4 or or & probbility = density Wht is when is not within,?
11 De. Rndom vribles X nd X with p.d.. nd respectively re independent i nd only i, = X nd X re independent. d The conditionl probbiliy density o rndom vrible X being given rndom vrible X being is Theorem 9: A nd B re independent events i nd only i P AB = PA P B, Similrly, the conditionl probbiliy density o rndom vrible X being given rndom vrible X being is De. The conditionl probbiliy density o rndom vrible X being given tht rndom vrible X being is, i.,
12 De. The joint probbility density unction o rndom vribles X nd X is unction cto, tht possesses the properties es i, ii, d d iii P X b, X b, d d b b De. The joint probbility density unction o rndom vribles X, X, i X is unction,,, tht possesses the properties,,..., ii...,,..., d d... d iii P X b, X b,..., X b b b b...,,..., d d... d
13 De. The mrginl probbiliy density unctions o rndom vribles X nd X,, d d &. re 3 De. The joint probbility density unction o rndom vribles X, X, X is unction,,, tht possesses the properties i,,...,, ii...,,..., d d... d b b b iii P X b, X b,..., X b...,,..., d d... d Given the joint probbility density unction o rndom vribles, the probbiliy density o the i th rndom vribles is i i...,,..., d d d i- d i+.. d In this contet, t the unction i i is clled the mrginl probbility bilit density o the i th rndom vrible X i.
14 De. Rndom vribles X nd X with p.d.. nd respectively 4 re independent i nd only i, = De. Rndom vribles X, X, X with p.d..,,..., respectively re independent i nd only i,,, = or ll vlues,,, o these rndom vribles. De. Rndom vribles X, X, X with cumultive distribution unctions F, F,..., F respectively re independent i nd only i F,,, = F F F or ll vlues,,, o these rndom vribles.
15 5 Consider unction gx o rndom vribles X. p is n oven temperture in degrees centigrde, then gx = 9X/5 + 3 Is the sme temperture in degrees Fhrenheit. For emple, i X The epected vlue o the unction gx is E[ gx ] = g d The epected vlue o X is E[ X ] = men. Men o probbility density d The epected vlue o X is E[ X ] = vrince. Vrince o probbility density d
16 The epected vlue o the unction gx is E[ gx X ] = g d eg e.g. Consider gx= = X+b b where nd b re ny constnts. Find the men nd vrince o gx. The epected vlue o X is E[ X ] men. The epected vlue o X is E[ X ] = vrince. 6 sln. The epected vlue o X + b is E[ X + b ] = E[X ] + b The vrince o X + b denoted s VrX + b is E[ X X+b b - E[X ] - b ] = E[ X X - ] = E[ X - ] =
17 e.g. Let X. hs men nd stndrd devition. Use the property o epecttion to show tht the stndrdized rndom vrible Z X hs men nd stndrd devition. 7 Sln. Let Z = X + b where = / nd b = - / The men o Z. = E[X ] + b = / + - / = The stndrd devition o Z. = rndom vrible X + b Men E[X+b] = + b Vrince VrX + b = = / =
18 8 De. The joint probbility density unction o rndom vribles X, X, X is unction i,,, tht possesses the properties,,..., ii...,,..., d d... d iii P X b, X b,..., X b...,,..., d d... d b b b The epected vlue o gx is E[ gx ] = g d The epected vlue o g X, X, X... g,,...,,,..., d d... d = E[ gx, X, X ] e.g. When gx, X = X - X - E[ X - X - ] is clled the covrince o X nd X.
19 9 De. The moments o joint p.d.., re i j i j i, j E[ X X ], d d - where i, j =,,, 3, nd the order o i,j is i+ j. X d, E [ ] men o X, d d - -, E[ X ] men o X d - -, d d De. The moments bout the men o joint p.d.., re i j i j i, j E X X ], - [ d d where i, j =,,, 3, nd dthe order o i,j is i+j. Note:, = E[ X - X - ] is the covrince o X nd X.
20 Find: Given rndom vribles X nd X, ind the three nd order moments bout the men. Sln: The three nd order moments bout the men re,,, nd,. De. The moments bout the men o joint p.d.., re i i, j E[ X X j ] μ, E[ X ] E [ X X ] E μ, E [ X ] E [ X X ] [ X ] E[ X] E [ X ] E [ X ] E[ X] E[ X ] μ, covrince E[ X E E X ] [ X X X X ] [ ] E X X ] E [ X ] E [ X ] [ ] E[X X ]
21 De. The numericl mesure o the similrity between X nd X is the normlised correltion coeicients nd is deined s. E, ] [ X X Note: [-, ] y e.g. Vrince o probbility density y d y y De: Rndom vribles X nd X re uncorrelted i = or, =.
22 Independent nd Uncorrelted Th m : I X nd X re sttisticlly i independent d then they re uncorrelted. Proo:, X ], d d E[ X X E [ X X ] E[ X d d d d De: ] De: X nd X re sttisticlly independent i, =. i.e. = nd so X nd X re uncorrelted. De: X nd X re uncorrelted i = or, =.
23 e.g. Let Y = X + X where X & X re independent rndom vribles, nd re ny constnts. Find Y = E[ Y ] nd Vr Y. 3 Sln. Y = E[ Y ] E[ [ gx, X, X ] = = E[ X + X ]...,.,,., d g... d, d d d d De: X nd X re sttisticlly independent i, =. d d d d d d d d E[ X] E[ X ]
24 4 e.g. continued Let Y = X + X where X & X re independent rndom vribles, nd re ny constnts. Find Y = E[ Y ] nd Vr Y. Sln. VrY = E[Y - Y ] = E[ X + X X X ] Vrince E[ X d Y = X + X = E[ X ]+ E[ X ] = E[ + X X X X ] Y E[ X ]+ E[ X ] = E[ [ X + X X X + X X X X ] = E[X X ] + E[X X ] + E[X X X X ] = VrX + VrX + Covrince, E X X ] = VrX + VrX [ X & X re independent they re uncorrelted. ]
25 Let Y = X + X where X & X re independent rndom vribles, nd re ny constnts. y Then Y = E[ Y ] = E[X ] + E[X ] nd Y = VrY = E[ Y - Y ] = VrX + VrX 5 Let Y = X + X + + X where X, X,, X re independent rndom vribles,,, nd re ny constnts. Then Y = E[ [ Y ] = E[X[ ] + E[X[ ] + + E[X[ ] ] i i i Y = VrY = E[ Y - Y ] = VrX + VrX + + VrX i i i
26 Joint p.d.. o discrete rndom vribles 6 De. The joint probbility distribution o discrete rndom vribles X nd X is unction, tht possesses the properties i, ii, vlues vlues iii P X b, X b [, b ] [, b ], De. The mrginl probbiliy distributions o discrete rndom vribles X nd X re respectively, vlues vlues, & The joint probbility distribution o n discrete rndom vribles nd the corresponding mginl probbility distributions cn be deined in similr wy.
27 De. The conditionl probbiliy density o continuous rndom vrible X being given tht rndom vrible X being, provided. is 7 De. The conditionl probbiliy distribution o discrete rndom vrible X being given tht rndom vrible X being, provided. e.g. Find rom, is the given, = =6 =.6 Sln = =.4 =./.6= /6.4/.6= 4/6./.6= /6 =./.4= ½./.4= ½ /.4=
28 8 De. Rndom vribles X, X, X with p.d..,,..., respectively re independent i nd only i,,, = or ll vlues,,, o these rndom vribles. De. Rndom vribles X, X, X with cumultive distribution unctions F, F,..., F respectively re independent d i nd only i F,,, = F F F or ll vlues,,, o these rndom vribles. The deinitions o n independent rndom vribles cn be pplied to both continuous nd discrete rndom vribles without modiiction.
29 Given rndom vribles X, X, X, which cn be discrete or continuous, the unction Y = g X, X, X is lso rndom vrible. We cn ind its epected vlue. 9 In the continuous cse, E[ Y ] = E[ gx, X, X ]... g,,...,,,..., d d... d In the discrete cse, E[ Y ] = E[ gx, X, X ]... g,,...,,,...,
30 3 Suppose Y = X + X + + X where X, X,, X re independent rndom vribles,,, nd Then Y = E[ Y ] = E[X ]+ E[X ] + + E[X ] i i i re ny constnts. V + Y = VrY = E[ Y - Y ] = VrX VrX + + VrX i i i The results bove cn be pplied to both continuous nd discrete rndom vribles without modiiction.
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