Continuous Random Variables: Joint PDFs, Conditioning, Expectation and Independence

Transcription

1 Cotiuous Radom Variables: Joit DFs, Coditioig, xpectatio ad Idepedece Berli Che Departmet o Computer ciece & Iormatio gieerig Natioal Taiwa Normal Uiversit Reerece: - D.. Bertsekas, J. N. Tsitsiklis, Itroductio to robabilit, ectios

2 Multiple Cotiuous Radom Variables (/) Two cotiuous radom variables ad associated with a commo experimet are joitl cotiuous ad ca be described i terms o a joit DF satisig, B x, is a oegative uctio Normalizatio robabilit, x, dxd, a, c imilarl, ca be viewed as the probabilit per uit area i the viciit o a, c a a a c c a Where is a small positive umber x, B,,, dxd, c c x, dxd a c,,, robabilit-berli Che

3 Multiple Cotiuous Radom Variables (/) Margial robabilit ad, x, ddx We have alread deied that We thus have the margial DF imilarl, x x x,, x,, d dx dx robabilit-berli Che 3

4 Illustrative xample xample 3.0. Two-Dimesioal Uiorm DF. We are told that the joit DF o the radom variables ad is a costat c o a area ad is zero outside. Fid the value o c ad the margial DFs o ad. The correspod ig uiorm joit DF o a area is deied to be (c. xample 3.9), x,, ize 0, 4, i x, o area otherwise x, or x, or 4 x x, or x 3 x 3 x x, 4 3, 4, 4 d d d d or or x, 3 x, 3 3 4, 4, dx dx 4 dx dx or 3 x, 4, 4 dx 4 dx robabilit-berli Che 4

5 Joit CDFs I ad are two (either cotiuous or discrete) radom variables associated with the same experimet, their joit cumulative distributio uctio (Joit CDF) is deied b F I ad urther have a joit DF ( ad are cotiuous radom variables), the d x, x,,,, x,, x s t F,, I ca be dieretiated at the poit F x,, x, x F, x, dsdt robabilit-berli Che 5

6 Illustrative xample xample 3.. Veri that i ad are described b a uiorm DF o the uit square, the the joit CDF is give b F x, x, x, or 0 x,, 0,, 0,0,0 F, x x,, x,, or all x, i the uit square robabilit-berli Che 6

7 xpectatio o a Fuctio o Radom Variables I ad are joitl cotiuous radom variables, ad g is some uctio, the Z g, is also a radom variable (ca be cotiuous or discrete) The expectatio o Z ca be calculated b Z g gx, x,,, dxd Z b I is a liear uctio o ad, e.g., Z a, the Z a b a b a Where ad are scalars b We will see i ectio 4. methods or computig the DF o Z (i it has oe). robabilit-berli Che 7

8 More tha Two Radom Variables The joit DF o three radom variables, ad Z is deied i aalog with the case o two radom variables,, Z B x, z The correspodig margial probabilities The expected value rule takes the orm g I is liear (o the orm a b cz ), the,, Z,, Z, B x,,, Z x,, z x x,, z,,, Z dz ddz dxddz g, Z gx,, z x, z,,, a b cz a b cz Z, dxd dz robabilit-berli Che 8

9 Coditioig DF Give a vet (/3) The coditioal DF o a cotiuous radom variable, give a evet I caot be described i terms o, the coditioal DF is deied as a oegative uctio satisig B xdx B x Normalizatio propert x dx robabilit-berli Che 9

10 Coditioig DF Give a vet (/3) I ca be described i terms o ( is a subset o the real lie with 0 ), the coditioal DF is deied as a oegative uctio x satisig x 0, The coditioal DF is zero outside the coditioig evet ad or a subset x Normalizatio ropert, i x otherwise B B, B B B x dx x dx x dx x dx remais the same shape as except that it is scaled alog the vertical axis robabilit-berli Che 0

11 Coditioig DF Give a vet (3/3),,, I are disjoit evets with i 0 or each i, that orm a partitio o the sample space, the x x i i i Veriicatio o the above total probabilit theorem x x x i x t dt t x x i i i i i Takig the derivative o both sides with respect to x i i i dt thik o x as a evet B, ad use the total probabilit theorem rom Chapter robabilit-berli Che

12 Illustrative xamples (/) t xample 3.3. The expoetial radom variable is memorless. T is expoetial T t t e T e 0, The time T util a ew light bulb burs out is expoetial distributio. Joh turs the light o, leave the room, ad whe he returs, t time uits later, id that the light bulb is still o, which correspods to the evet ={T>t} Let be the additioal time util the light bulb burs out. What is the coditioal DF o give? T t, T t t, t 0 otherwise The coditioal CDF o x T t x T t (where x T t x ad T t T t xt t T t x T t et x et ex give is deied b T t 0) The coditioa l DF o the evet is also expoetia l with parameter. give robabilit-berli Che

13 Illustrative xamples (/) /0 xample 3.4. The metro trai arrives at the statio ear our home ever quarter hour startig at 6:00 M. ou walk ito the statio ever morig betwee 7:0 ad 7:30 M, with the time i this iterval beig a uiorm radom variable. What is the DF o the time ou have to wait or the irst trai to arrive? - The arrival time, deoted b, is a uiorm radom variableover theiterval 7 :0 to 7 : 30 - Let radom varible model the waitig time - Let be a evet 7 :0 7 :5(ou board the 7 :5 trai) - Let B be a evet B 7 :5 7 : 30(ou board the 7 : 30 trai) - Let be uiorm coditioed o - Let be uiorm coditioed o B Total robabilit theorem: B B For 0 For 5 5, 5, robabilit-berli Che 3

14 Coditioig oe Radom Variable o other Two cotiuous radom variables ad have a joit DF. For a with 0, the coditioal DF o give that is deied b Normalizatio ropert x, x, x dx The margial, joit ad coditioal DFs are related to each other b the ollowig ormulas x, x,, x x, d., margializatio robabilit-berli Che 4

15 Illustrative xamples (/) Notice that the coditioal DF x has the same shape as the joit DF, x,, because the ormalizig actor does ot deped o x Figure 3.6: Visualizatio o the coditioal DF. Let, have a joit DF which is uiorm o the set. For each ixed, we cosider the joit DF alog the slice ad ormalize it so that it itegrates to x 3.5 x 3.5 x 3.5 x, x,, x,3.5,.5 3.5,.5.5 x.5 c. example 3.3 / 4 / 4 / 4 / / / 4 / 4 robabilit-berli Che 5

16 Illustrative xamples (/) xample 3.5. Circular Uiorm DF. Be throws a dart at a circular target o radius r. We assume that he alwas hits the target, ad that all poits o impact x, are equall likel, so that the joit DF, x, o the radom variables x ad is uiorm What is the margial DF, x,, i x, is i the circle area o the circle 0, otherwise, x r πr 0, otherwise, x, dx x r dx πr r dx dx x r πr πr r r, i r πr (Notice here that DF is ot uiorm) x, x, πr r πr, i x r For each value, x is uiorm robabilit-berli Che 6 r

17 Coditioal xpectatio Give a vet The coditioal expectatio o a cotiuous radom variable, give a evet ( ), is deied b The coditioal expectatio o a uctio also has the orm g gx xdx x x dx 0 g Total xpectatio Theorem ad i i i g g i Where,,, are disjoit evets with i 0 each i, that orm a partitio o the sample space i i or robabilit-berli Che 7

18 Illustrative xample xample 3.7. Mea ad Variace o a iecewise Costat DF. uppose that the radom variable has the piecewise costat DF Deie x / 3, / 3, 0, lies i 0 x, i x, otherwise. evet i the irst iterval [0,] evet lies i the secod iterval 0 / 3dx / 3, / 3dx / 3 x, 0 x x x Recall that the mea ad secod momet o a uiorm radom variable over a iterval a b/ ad a ab b / 3 [ a, b ] is /, / 3 3 /, 7 / 3 0, otherwise var [,] 0, / 3 / / 3 3 / x / 3 / 3 / 3 7 / 3 5 / 9 5 / 9 7 / 6 / 36, x otherwise 7 / 6 robabilit-berli Che 8

19 robabilit-berli Che 9 Coditioal xpectatio Give a Radom Variable The properties o ucoditioal expectatio carr though, with the obvious modiicatios, to coditioal expectatio dx x x dx x x g g dx x x g g,,

20 Total robabilit/xpectatio Theorems Total robabilit Theorem For a evet ad a cotiuous radom variable d Total xpectatio Theorem For a cotiuous radom variables ad g g g, g, d d d robabilit-berli Che 0

21 Idepedece Two cotiuous radom variables ad are idepedet i ice that x, x, or all x,,, x, x x x We thereore have x x or all x ad all with, 0 Or x or all ad all x with x, 0 robabilit-berli Che

22 More Factors about Idepedece (/) I two cotiuous radom variables ad are idepedet, the two evets o the orms ad B are F, idepedet It also implies that, B x B, x x B x x xdx B B, ddx ddx x, x, x F xf x The coverse statemet is also true (ee the ed-o-chapter problem 8) d robabilit-berli Che

23 More Factors about Idepedece (/) I two cotiuous radom variables ad are idepedet, the var var var The radom variables g ad h are idepedet or a uctios g ad h Thereore, g h g h robabilit-berli Che 3

24 Recall: the Discrete Baes Rule Let,,, be disjoit evets that orm a partitio o the sample space, ad assume that i 0, or all i. The, or a evet such that we have B B 0 i B B i B i i B i k k B k i B i B B Multiplicatio rule Total probabilit theorem robabilit-berli Che 4

25 Ierece ad the Cotiuous Baes Rule s we have a model o a uderlig but uobserved pheomeo, represeted b a radom variable with DF, ad we make a ois measuremet, which is modeled i terms o a coditioal DF. Oce the experimetal value o is measured, what iormatio does this provide o the ukow value o? x Measuremet Ierece x x x, x, x x t t dt Note that, robabilit-berli Che 5

26 robabilit-berli Che 6 Ierece ad the Cotiuous Baes Rule (/) I the uobserved pheomeo is iheretl discrete Let is a discrete radom variable o the orm that represets the dieret discrete probabilities or the uobserved pheomeo o iterest, ad be the MF o N N N p N i N N N N N N i i p p p N N N N Total probabilit theorem Ierece about a Discrete Radom Variable

27 Illustrative xamples (/) xample 3.9. lightbulb produced b the Geeral Illumiatio Compa is kow to have a expoetiall distributed lietime. However, the compa has bee experiecig qualit cotrol problems. O a give da, the parameter o the DF o is actuall a radom variable, uiorml distributed i the iterval, 3 /. I we test a lightbulb ad record its lietime ( ), what ca we sa about the uderlig parameter? e, 0, 0, or 3 / 0, otherwise 3 / t t dt Coditioed o, has a expoetial distributio with parameter 3 / e te t, or λ 3/ dt robabilit-berli Che 7

28 robabilit-berli Che 8 Illustrative xamples (/) xample 3.0. igal Detectio. biar sigal is trasmitted, ad we are give that ad. The received sigal is, where is a ormal oise with zero mea ad uit variace, idepedet o. What is the probabilit that, as a uctio o the observed value o? p p N N - - s e s s ad, ad or, / Coditioed o, has a ormal distributio with mea ad uit variace s s e p pe pe e p e pe e pe e e p e p e p p p p p / / / / / /

29 Ierece Based o a Discrete Radom Variable The earlier ormula expressig i terms o ca be tured aroud to ield t t dt? d d ( ormalizat io propert : d ) d robabilit-berli Che 9

30 Recitatio CTION 3.4 Joit DFs o Multiple Radom Variables roblems 5, 6 CTION 3.5 Coditioig roblems 8, 0, 3, 4 CTION 3.6 The Cotiuous Baes Rule roblems 34, 35 robabilit-berli Che 30