Cotuous Radom Varables: Codtog, xectato ad Ideedece Berl Che Deartmet o Comuter cece & Iormato geerg atoal Tawa ormal Uverst Reerece: - D.. Bertsekas, J.. Tstskls, Itroducto to robablt, ectos 3.4-3.5
Codtog DF Gve a vet (/3) The codtoal DF o a cotuous radom varable, gve a evet I caot be descrbed terms o, the codtoal DF s deed as a oegatve ucto satsg ( B ) ( x)dx B ( x) ormalzato roert ( x ) dx robablt-berl Che
Codtog DF Gve a vet (/3) I ca be descrbed terms o ( s a subset o the real le wth ( ) > 0 ), the codtoal DF s deed as a oegatve ucto x satsg ( x ) 0, ad or a subset ( x ) ( ) ( B ) B ( ), x otherwse ( B, ) ( ) I B ( x ) dx ( ) ( x ) dx B remas the same shae as excet that t s scaled alog the vertcal axs ormalzato roert ( x ) dx ( x ) dx robablt-berl Che 3
Codtog DF Gve a vet (3/3),,, I K are dsjot evets wth ( ) > 0 or each, that orm a artto o the samle sace, the ( x ) ( ) ( x ) Vercato o the above total robablt theorem ( x) ( ) ( x ) x Takg the x () t dt ( ) () t ( x) ( ) ( x) dervatve dt o both sdes wth resectve to x robablt-berl Che 4
Codtoal xectato Gve a vet The codtoal exectato o a cotuous radom varable, gve a evet ( > ), s deed b The codtoal exectato o a ucto g also has the orm Total xectato Theorem Where,, K, are dsjot evets wth ( ) > 0 each, that orm a artto o the samle sace [ ] x ( x ) dx [ g( ) ] g( x) ( x) [ g( )] ( ) g( ) ( ) 0 ( ) [ ] dx or robablt-berl Che 5
Illustratve xamle xamle 3.0. Mea ad Varace o a ecewse Costat DF. uose that the radom varable has the ecewse costat DF / 3, 0 x, ( x) / 3, x, 0, otherwse. Dee evet { les the rst terval [0,] } evet { les the secod terval [,] } ( ) / 3dx / 3, ( ) / 3dx / 3 0 Recall that the mea ad secod momet o a uorm radom varable over a terval ( a + b) / ad ( a + ab b )/ 3 [ a, b ] s + [ ] /, [ ] / 3 [ ] 3 /, [ ] 7 / 3 var [ ] ( ) [ ] + ( ) [ ] / 3 / + / 3 3 / [ ] ( ) [ ] + ( ) [ ] / 3 / 3 + / 3 7 / 3 5 / 9 ( ) 5 / 9 ( 7 / 6) / 36 7 / 6 robablt-berl Che 6
Multle Cotuous Radom Varables Two cotuous radom varables ad assocated wth a commo exermet are jotl cotuous ad ca be descrbed terms o a jot DF satsg ((, ) B) ( x ) ) B, s a oegatve ucto ormalzato robablt, ) dxd, ( a, c) mlarl, ca be vewed as the robablt er ut area the vct o ( a, c) Where δ s a small ostve umber,,, dxd ( a a + δ, c c + δ ) a+ δ c+ δ ) dxd ( a c) δ a c,,, robablt-berl Che 7
Illustratve xamle xamle 3.3. Two-Dmesoal Uorm DF. We are told that the jot DF o the radom varables ad s a costat c o a area ad s zero outsde. Fd the value o c ad the margal DFs o ad. or The a area, corresod ), x g s deed uorm ze o area 0, 4 to be (c. xamle, jot ) or ) 4 ( x) ) 4 d 4 or x 3 3 ( x) ),, 3 4 3 d 4 4 d d DF ) o otherwse 3.) or or ( ) ) dx 4 3 3 ( ) ),, 4 3 dx 4 dx dx or 3 ( ) ) 4, dx 4 4 dx robablt-berl Che 8
Codtog oe Radom Varable o other Two cotuous radom varables ad have a jot DF. For a wth ( ) > 0, the codtoal DF o gve that s deed b ( x ), ) ( ) ormalzato roert ( x ) dx The margal, jot ad codtoal DFs are related to each other b the ollowg ormulas ) ( ) ( x ),, ( x) ) d., margalzato robablt-berl Che 9
Illustratve xamles (/) ( ) otce that the codtoal DF x has the same shae as the jot DF, ), because the ormalzg actor ( ) does ot deed o x ( x 3.5), 3.5) ( 3.5).5) (.5),.5 ) (.5) ( ) Fgure 3.7: Vsualzato o the codtoal DF x. Let, have a jot DF whch s uorm o the set. For each xed, we cosder the jot DF alog the slce ad ormalze t so that t tegrates to ( x 3.5) ( x 3.5), c. examle 3.3 / 4 / 4 / 4 / / / 4 / 4 robablt-berl Che 0
Illustratve xamles (/) xamle 3.5. Crcular Uorm DF. Be throws a dart at a crcular target o radus r. We assume that he alwas hts the target, ad that all ots o mact ) are equall lkel, so that the jot DF, ) o the radom varables x ad s uorm What s the margal DF ( ), ), area o the crcle 0,, x + r πr 0, otherwse ( ) ), dx x + x + r dx πr πr r, r πr (otce here that DF r ) otherwse πr ( ) s ot uorm) r r dx s the crcle dx ( x ), ) ( ) πr r πr, x r For each value, x s uorm robablt-berl Che ( ) + r
xectato o a Fucto o Radom Varables I ad are jotl cotuous radom varables, ad g s some ucto, the Z g(, ) s also a radom varable (ca be cotuous or dscrete) The exectato o Z ca be calculated b [ Z ] [ g( )] g ) ( x ),,, dxd Z b I s a lear ucto o ad, e.g., Z a +, the [ Z ] [ a + b ] a[ ] b[ ] + a b Where ad are scalars robablt-berl Che
robablt-berl Che 3 Codtoal xectato The roertes o ucodtoal exectato carr though, wth the obvous modcatos, to codtoal exectato [ ] ( ) dx x x ( ) [ ] ( ) ( ) dx x x g g ( ) [ ] ( ) ( ) dx x x g g,,
Total robablt/xectato Theorems Total robablt Theorem For a evet ad a cotuous radom varable ( ) ( ) ( ) d Total xectato Theorem For a cotuous radom varables ad [ ] [ ] ( ) [ g ( )] g ( ) d [ ] ( ) [ g (, )] g (, ) d [ ] ( ) d robablt-berl Che 4
Ideedece Two cotuous radom varables ad are deedet ) ( x) ( ), or all x,, Or ( x ) ( x) or all x ad all wth ( ), > 0 Or ( x) ( ) or all ad all x wth ( x), > 0 robablt-berl Che 5
More Factors about Ideedece (/) I two cotuous radom varables ad are deedet, the two evets o the orms ad B are deedet { } { } (, B) x B, ( x ) x B ( x) ( ) x ( x) dx B ( )( B), ddx ddx [ ] d [ ] ( ) robablt-berl Che 6
More Factors about Ideedece (/) I two cotuous radom varables ad are deedet, the [ ] [ ] [ ] ( + ) var ( ) var ( ) var + The radom varables ad h are deedet or a uctos g ad h Thereore, g ( ) ( ) [ g( ) h( )] [ g( )] [ h( )] robablt-berl Che 7
Jot CDFs I ad are two (ether cotuous or dscrete) radom varables, ther jot cumulatve dstrbuto ucto (CDF) s deed b F I ad urther have a jot DF, the d ) ( x, ),,, ( x ) ( s t) F,,,, ) x F, x ) I ca be deretated at the ot dsdt F, ) robablt-berl Che 8
Illustratve xamle xamle 3.0. Ver that ad are descrbed b a uorm DF o the ut square, the the jot CDF s gve b F ) ( x, ) x, or 0 x,, ( 0,) (, ) ( 0,0) (,0) F, x ), ), or all ) the ut square robablt-berl Che 9
Recall: the Dscrete Baes Rule Let,, K, be dsjot evets that orm a artto o the samle sace, ad assume that ( ) 0, or all. The, or a evet such that we have B ( B) > 0 ( B) ( ) ( B ) ( B) ( ) ( ) B ( ) ( ) k k B k ( ) ( ) B ( ) ( B ) + L + ( ) ( B ) Multlcato rule Total robablt theorem robablt-berl Che 0
Ierece ad the Cotuous Baes Rule (/) s we have a model o a uderlg but uobserved heomeo, rereseted b a radom varable wth DF, ad we make a os measuremet, whch s modeled terms o a codtoal DF. Oce the exermetal value o s measured, what ormato does ths rovde o the ukow value o? ( x) Measuremet Ierece ( ) x ( x ) ( x ), ) ( ) ( x) ( ) x () t ( t) dt robablt-berl Che
robablt-berl Che Ierece ad the Cotuous Baes Rule (/) I the uobserved heomeo s heretl dscrete Let s a dscrete radom varable o the orm that reresets the deret dscrete robabltes or the uobserved heomeo o terest, ad be the MF o { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) + + + δ δ δ δ δ Total robablt theorem
Illustratve xamles (/) xamle 3.8. lghtbulb roduced b the Geeral Illumato Coma s kow to have a exoetall dstrbuted letme. However, the coma has bee exerecg qualt cotrol roblems. O a gve da, the arameter Λ λ o the DF o s actuall a radom varable, uorml dstrbuted the terval [, 3 / ]. I we test a lghtbulb ad record ts letme ( ), what ca we sa about the uderlg arameter λ? Λ Λ Λ λ ( λ ) λe, 0, λ > 0 ( λ ), or λ 3 / 0, otherwse ( λ ) 3 / Λ ( λ ) Λ ( λ ) () t ( t) Λ Λ dt Codtoed o Λ λ, has a exoetal dstrbuto wth arameter λ 3 / λe te λ t, or λ 3/ dt robablt-berl Che 3
robablt-berl Che 4 Illustratve xamles (/) xamle 3.9. gal Detecto. bar sgal s trasmtted, ad we are gve that ad. The receved sgal s, where ormal ose wth zero mea ad ut varace, deedet o. What s the robablt that, as a ucto o the observed value o? ( ) ( ) + ( ) ( ) - - s e s s ad, ad or, / π σ Codtoed o, has a ormal dstrbuto wth mea ad ut varace s s ( ) ( ) ( ) ( ) () ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e e e e e e e e e e e e + + + + + + + + / / / / / / π π π
Rectato CTIO 3.4 Codtog o a vet roblems 4, 7, 8 CTIO 3.5 Multle Cotuous Radom Varables roblems 9, 4, 5, 6, 8 robablt-berl Che 5