Pairs of Random Variables

Transcription

1 LECTURE 4 Pairs of Random Variables 4. TWO RANDOM VARIABLES Let(,F, P)beaprobabl spaceandconsidertwomeasurablemappings X : Ă Ă, Y : Ă Ă. In other words, X andy are two random variables defined on the common probabilit space(,f, P). Inordertospeciftherandomvariables, weneedtodetermine P{(X,Y) A} = P({ω : (X(ω),Y(ω)) A}) foreverborel set A Ă 2. B thepropertiesof theborel σ-field, itcan be shownthatit suffices to determine the probabilities of the form or equivalentl, the probabilities of the form The latter defines their joint cdf P{a < X < b, c < Y < d}, a < b, c < d, P{X, Y },, Ă. F X,Y (, ) = P{X, Y },, Ă, whichistheshadedregioninfigure.. The joint cdf satisfies the following properties:. F X,Y (, ) 0.. If 2 and 2,then. Limits. F X,Y (, ) F X,Y ( 2, 2 ). lim F,Ă X,Y(, ) =, lim Ă F X,Y(, ) = lim Ă F X,Y(, ) = 0.

2 2 Pairs of Random Variables (, ) Figure.. Anillustrationofthejointpdfof X andy.. Marginal cdfs. lim F Ă X,Y(, ) = F X (), lim F Ă X,Y(, ) = F Y (). Theprobabilit ofan(borel)setcanbedeterminedfromthejointcdf. Foreample, P{a < X b, c < Y d} = F X,Y (b, d) F X,Y (a,d) F X,Y (b, c)+f X,Y (a, c) as illustrated in Figure. Wesa that X andy are statisticallindependent or independent in shortif forever d c a b Figure.. Anillustrationof P{a < X b, c <Y d}.

3 4.2 Pairs of Discrete Random Variables 3 (Borel) AandB, P{X A, Y B} = P{X A} P{Y B}. Equivalentl, X and Y are independent if F X,Y (, ) = F X ()F Y (),, Ă. In the following, we focus on three special cases and discuss the joint, marginal, and conditional distributions of X and Y for each case. Ȃ X andy arediscrete. Ȃ X andy arecontinuous. Ȃ X isdiscreteandy iscontinuous (mied). 4.2 PAIRS OF DISCRETE RANDOM VARIABLES Let X andy be discrete randomvariables onthesame probabilit space. Theare completel specified b their joint pmf B the aioms of probabilit, p X,Y (, ) = P{X =,Y = }, X, Y. X p X,Y (, ) =. Y Weusethelawoftotalprobabilit tofindthemarginalpmf of X: p X () = p(, ), X. Y Theconditionalpmf of X giveny = isdefinedas p X Y ( ) = p X,Y(, ), p p Y () Y () = 0, X. Checkthatif p Y () = 0,then p X Y ( )isapmffor X. Eample.. Consider the pmf p(, ) described b the following table

4 4 Pairs of Random Variables Then, and /4 = 0, p X () = 3/8 =, 3/8 = 2.5, /2 = 0, p X Y ( 2) = /2 =. Chainrule. p X,Y (, ) = p X ()p Y X ( ) = p Y ()p X Y ( ). Independence. X and Y are independent if which is equivalent to Lawoftotalprobabilit.Foranevent A, p X,Y (, ) = p X ()p Y (), X, Y, p X Y ( ) = p X (), X, p Y () = 0. P(A) = p X () P(A X = ). Baesrule. Given p X () and p Y X ( ) for ever (, ) X Y, we can find p X Y ( ) as p X Y ( ) = p X,Y(, ) p Y () = p X()p Y X ( ) p Y () = p X()p Y X ( ) u X p X,Y (u, ) p X ()p Y X ( ) = p X (u)p Y X ( u). u X Thefinalformulaisentirelintermsoftheknownquantities p X ()and p Y X ( ). Eample. (Binar smmetric channel). Consider the binar communication channelinfigure.. Thebitsentis X Bern(p), p [0,],andthebitreceived is Y = (X + Z) mod 2 = X Z, wherethenoisez Bern(є), є [0,/2],isindependentofX. Wefind p X Y ( ), p Y (),

5 4.2 Pairs of Discrete Random Variables 5 Z {0,} X {0,} Y {0,} Figure.. Binar smmetric channel. and P{X = Y},namel,theprobabilit oferror. First,weusetheBaesrule p Y X ( ) p X Y ( ) = p Y X ( u)p X (u) p X(). u X Weknow p X (),butweneedtofind p Y X ( ): Therefore p Y X ( ) = P{Y = X = } = P{X Z = X = } = P{ Z = X = } = P{Z = X = } = P{Z = } since Z and X areindependent = p Z ( ). p Y X (0 0) = p Z (0 0) = p Z (0) = є, p Y X (0 ) = p Z (0 ) = p Z () = є, p Y X ( 0) = p Z ( 0) = p Z () = є, p Y X ( ) = p Z ( ) = p Z (0) = є. Plugging into the Baes rule equation, we obtain p X Y (0 0) = p Y X (0 0) p Y X (0 0)p X (0)+ p Y X (0 )p X () p X(0) = єp p X Y ( 0) = p X Y (0 0) = ( є)( p)+ єp. p X Y (0 ) = p Y X ( 0) p Y X ( 0)p X (0)+ p Y X ( )p X () p X(0) = p X Y ( ) = p X Y (0 ) = Wealreadfound p Y ()as ( є)p ( є)p+є( p). p Y () = p Y X ( 0)p X (0)+ p Y X ( )p X () = ( є)( p)+ єp for = 0, є( p)+( є)p for =. ( є)( p) ( є)( p)+єp. є( p) ( є)p+є( p).

6 6 Pairs of Random Variables Nowtofindtheprobabilit oferror P{X = Y},consider P{X = Y} = p X,Y (0,)+ p X,Y (,0) = p Y X ( 0)p X (0)+ p Y X (0 )p X () = є( p)+ єp = є. Alternativel, P{X = Y} = P{Z = } = є. An interesting special case is є = /2, whence P{X = Y} = /2,whichistheworstpossible(noinformationissent),and p Y (0) = 2 p+ 2 ( p) = 2 = p Y(). ThereforeY Bern(/2),regardlessofthevalue of p. Inthiscase,thebitsent X andthe bit received Y are independent(check!). 4.3 PAIRS OF CONTINUOUS RANDOM VARIABLES 4.3. Joint and Marginal Densities WesathatXandY arejointlcontinuousrandomvariablesiftheirjointcdfiscontinuous in both and. In thiscase, we can define theirjoint pdf, provided that it eists, as the function f X,Y (, )suchthat F X,Y (, ) = IfF X,Y (, )isdifferentiable in and,then f X,Y (u, )dud,, Ă. f X,Y (, ) = 2 F(, ) F X,Y ( +, + ) F X,Y ( +, ) F X,Y (, + )+F X,Y (, ) = lim, Ă0 P{ < X +, < Y + } = lim., Ă0 Thejointpdf f X,Y (, )satisfiesthefollowingproperties:. f X,Y (, ) 0.. f X,Y (, )d d =.. TheprobabilitofansetA Ă 2 canbecalculatedbintegratingthejointpdfovera: P{(X,Y) A} = f X,Y (, )d d. (,) A

7 4.3 Pairs of Continuous Random Variables 7 The marginalpdf of X can be obtained from thejoint pdf via thelaw of total probabilit: f X () = f X,Y (, )d. Toseethis,recallthat and consider д() = d d F X () = lim F Ă X,Y (, ) = д(u) du f X,Y (u, )ddu with д(u) = f X,Y(u, )d. We then differentiate F X () with respect to. Alternativel, we can consider the following figure + and note that P{ < X + } f X () = lim Ă0 = lim Ă0 lim Ă0 = lim Ă0 = n= f X,Y (, )d. P{ < X +, n < Y (n+) } f X,Y (, )d Eample.. Let(X,Y) f(, ),where c 0, 0, +, f(, ) = 0 otherwise. Wefind c, f Y (),and P{Y 2X}. Notefirstthat = f(, )d d = 0 0 cd d = c ( )d = 2 c. 0

8 8 Pairs of Random Variables Hence, c = 2. Tofind f Y (),weusethelawoftotalprobabilit f Y () = f(, )d = ( ) 2d 0, 0 0 otherwise 2( ) 0, = 0 otherwise. f Y () Tofindtheprobabilit oftheset{y 2X},wefirstsketchtheset 2 3 = 2 Fromthefigurewefindthat P X 2 Y = {(,): 2 } f X,Y(, )d d = ( ) d d = RecallthatXandYareindependentif f X,Y (, )= f X ()f Y ()forever,.but f X,Y (0,)= 2and f X (0)f Y () = 0,so X andy arenot independent.alternativel, consider toseethat X andy aredependent. f X,Y (/2,/2) = 2 = = f X (/2)f Y (/2)

9 4.3 Pairs of Continuous Random Variables Conditional Densities LetX andy becontinuousrandomvariableswithjointpdf f X,Y (, ). Wewishtodefine F Y X ( X = ) = P{Y X = }. We cannot define the above conditional probabilit as P{Y, X = } P{X = } since both the numerator and the denominator are equal to zero. Instead, we define the conditional probabilit for continuous random variables as a limit F Y X ( ) = lim Ă0 P{Y < X + } P{ < X +, } = lim Ă0 P{ < X + } + Ă0 = lim Ă0 = lim = f X,Y (u, )dud f X () f X,Y(, ) d f X () f X,Y (, ) d. f X () B differentiating w.r.t., we thus define the conditional pdf as f Y X ( ) = f X,Y(, ) f X () if f X () = 0. Itcanbereadil checkedthat f Y X ( )isavalid pdfand F Y X ( ) = f Y X (u )du isavalidcdfforever suchthat f X () > 0. Sometimesweusethenotation Y {X = } f Y X ( ) todenotey hasaconditionalpdf f Y X ( )given X =. Chainrule. f X,Y (, ) = f X ()f Y X ( ) = f Y ()f X Y ( ). Independence. X and Y are independent iff f X,Y (, ) = f X ()f Y (),, Ă, or equivalentl, f X Y ( ) = f X (), Ă, f Y () = 0.

10 0 Pairs of Random Variables Lawoftotalprobabilit.Foranevent A, P(A) = f X () P(A X = )d. Baesrule. Given f X ()and f Y X ( ),wecanfind f X Y ( ) = f Y X( ) f Y () f X () = f X()f Y X ( ) f X,Y(u, )du = f X ()f Y X ( ) f X(u)f Y X ( u)du. Eample.. AsinEample.,let f(, )bedefinedas 2 0, 0, +, f(, ) = 0 otherwise. We alread know that 2( ) 0, f Y () = 0 otherwise. Therefore f X Y ( ) = f X,Y(, ) f Y () = 0 <, 0, 0 otherwise. Inotherwords, X {Y = } Unif[0, ]. f X Y ( )

11 4.4 Pairs of Mied Random Variables Eample.. LetΛ Unif[0,]andtheconditionalpdfof X given{λ = λ}be f X Λ ( λ) = λe λ, 0 < λ, i.e., X {Λ = λ} Ep(λ). Then,btheBaesrule, f Λ X (λ 3) = f X Λ (3 λ)f Λ (λ) 0 f Λ(u)f X Λ (3 u)du λe 3λ = 0 < λ, 9 ( 4e 3 ) 0 otherwise. f Λ (λ) 0 f Λ X (λ 3) λ λ 4.4 PAIRS OF MIXED RANDOM VARIABLES Let X be a discrete random variable withpmf p X (). For each with p X () = 0, conditionedontheevent{x = },lety beacontinuousrandomvariable withconditionalcdf F Y X ( )andconditionalpdf f Y X ( ) = F Y X( ), provided that the derivative is well-defined. Then, b the law of total probabilit, F Y () = P{Y } = P{Y X = }p X () = F Y X ( )p X () = f Y X ( )d p X (),

12 2 Pairs of Random Variables and hence f Y () = d d F Y() = f Y X ( )p X (). Theconditionalpmfof X giveny = canbedefinedasalimit: P{X =, < Y + } p X Y ( ) = lim Ă0 P{ < Y + } p = lim X () P{ < Y + X = } Ă0 P{ < Y + } p X ()f Y X ( ) = lim Ă0 f Y () = f Y X( ) p f Y () X (). Chainrule. p X ()f Y X ( ) = f Y ()p X Y ( ). Baesrule. Given p X ()and f Y X ( ), p X Y ( ) = p X()f Y X ( ) u p X (u)f Y X ( u). Eample. (Additive Gaussian noise channel). Consider the communication channel in Figure.. The signal transmitted is a binar random variable X = + P w.p. p, P w.p. p. The received signal, also called the observation, is Y = X + Z, where Z N(0,N) is additive noiseand independentof X. First note thaty {X = } Z N(0,N) X Y Figure.. Additive Gaussian noise channel.

13 4.5 Signal Detection 3 N(,N). Toseethis,consider P{Y X = } = P{X + Z X = } = P{Z X = } = P{Z }, which, b taking derivative w.r.t., implies that In other words, f Y X ( ) = f Z ( ) = ( ) 2 2πN e 2. Y {X = + P} N(+ P,N) and Y {X = P} N( P,N). Net b the law of total probabilit, Finall, b the Baes rule, f Y () = p X (+ P)f Y X ( + P)+ p X ( P)f Y X ( P) = p 2πN e 2 +( p) 2πN e ( ) 2 ( + ) 2 2. and p X Y (+ P ) = = p ( ) 2 2πN e 2 p e ( )2 ( p) 2 + e ( + )2 2 2πN 2πN pe pe +( p)e p X Y ( P ) = ( p)e pe +( p)e. 4.5 SIGNAL DETECTION Consider the general digital communication sstem depicted in Figure., where the signal sent is X p X (), X = {, 2,..., n }, and theobservation (received signal) is Y {X = } f Y X ( ). A detector is a mapping d : Ă Ă X that generates an estimate X = d(y) of the input signal X. We wish to find an optimal detector d () that minimizes the probabilit of error P e := P{ X = X} = P{d(Y) = X}.

14 4 Pairs of Random Variables X {, 2,..., n } nois Y channel detector X {, 2,..., n } f Y X ( ) d() Figure.. The general digital communication sstem. Supposethatthereisnoobservationandwewouldliketofindthebestguessd of X, that is, d = argmin d P{d = X}. Clearl, the best guess is the one with the largest probabilit, that is, d = argma p X (). (. ) Thissimple observation continues toholdwiththeobservationy =. Definethemaimum a posteriori probabilit(map) detector as d () = argma p X Y ( ). (. ) Then, b(. ) P{d () = X Y = } P{d() = X Y = } for an detector d(). Consequentl, P{d (Y) = X} = P{d () = X Y = } P{d() = X Y = } = P{d(Y) = X}, andthemapdetectorminimizes theprobabilit oferrorp e. WhenX isuniformon{, 2,..., n },thatis, p X ( ) = p X ( 2 ) = = p X ( n ) = /n, p X Y ( ) = p X()f Y X ( ) f Y () = f Y X( ) nf Y (), whichdependson onl throughthelikelihood f Y X ( )foragiven. Then,theMAP detection rule in(. ) simplifies as the maimum likelihood(ml) detection rule d () = argma f Y X ( ).

15 4.5 Signal Detection 5 Eample.. We revisit the additive Gaussian noise channel in Eample. with signal X = + P w.p. 2 P w.p. 2 independentnoise Z N(0,N), andoutputy = X + Z. TheMAPdetectoris d () = + P if p X Y (+ P ) > p X Y ( P ), P otherwise. Sincethetwosignalsareequall likel,themapdetectionrulereducestothemldetetion rule d () = + P if f Y X ( + P) > f Y X ( P), P otherwise. f( P) f( + P) P + P Fromthefigureabove andusingthegaussian pdf,themap/mldetectorreduces tothe minimum distance detector d () = + P ( P) 2 < ( ( P)) 2, P otherwise, which further simplifies to d () = + P > 0, P 0. Now the minimum probabilit of error is P e = P{d (Y) = X} = P{X = P} P{d (Y) = P X = P} + P{X = P} P{d (Y) = P X = P} = 2 P{Y 0 X = P}+ 2 P{Y > 0 X = P} = 2 P{Z P}+ 2 P{Z > P} = Q P N. Note that the probabilit of error is a decreasing function of P/N, which is often called the signal-to-noise ratio(snr).

16 6 Pairs of Random Variables 4.6 FUNCTIONS OF TWO RANDOM VARIABLES Let (X,Y) f(, ) and let д(, ) be a differentiable function. To find the pdf of Z = д(x,y),wefirstfindtheinverseimageof{z z}tocomputeitsprobabilitepressedas afunctionof z,i.e.,f Z (z),andthentakethederivative. Eample.. Suppose that X f X () andy f Y () are independent. Let Z = X +Y. Then F Z (z) = P{Z z} = P{X +Y z} = = = = Taking derivative w.r.t. z, we have P{X +Y z X = }f X ()d P{ +Y z X = }f X ()d P{Y z }f X ()d F Y (z )f X ()d. f Z (z) = f Y (z )f X ()d, which is the convolution of f X () and f Y (). For eample, if X N( X,σ 2 X ) and Y N( Y,σ 2 Y )areindependent,thenitcanbereadil checkedthat Z = X +Y N( X + Y,σ 2 X + σ2 Y ). A similar result also holds for the sum of two independent discrete random variables (replacing pdfs with pmfs and integrals with sums). For eample, if X Poisson(λ ) and Y Poisson(λ 2 ) are independent, then Z = X +Y Poisson(λ ) Poisson(λ 2 ) = Poisson(λ + λ 2 ). The propert that thesumof two independentrandom variables with the same distribution has the same distribution, which is obeed b Gaussian and Poisson random variables, is referred to as infinite divisibilit. For eample, a Poisson(λ) r.v. can bewrittenasthesumofannumberofindependentpoisson(λ i )r.v.s,aslongas i λ i = λ. Itissometimes easiertoworkwithcdfsfirsttofindthepdfofafunctionof X andy (especiall when д(, ) is not differentiable). Eample. (Minimumandmaimum). LetX f X ()andy f Y ()beindependent. Define U = ma{x, Y} and V = min{x, Y}.

17 Problems 7 TofindthepdfofU,wefirstfinditscdf F U (u) = P{U u} = P{X u, Y u} = F X (u)f Y (u). Using the product rule for derivatives, f U (u) = f X (u)f Y (u) + f Y (u)f X (u). NowtofindthepdfofV,consider P{V > } = P{X >, Y > } F V ( ) = ( F X ( ))( F Y ( )). Thus f V ( ) = f X ( )+ f Y ( ) f X ( )F Y ( ) f Y ( )F X ( ). Moregenerall,thejointpdfof(U,V)canbefoundbsimilararguments;seeProblem??. PROBLEMS.. Geometric with conditions. Let X be a geometric random variable with pmf p X (k) = p( p) k, k =,2,... Findandplottheconditionalpmf p X (k A) = P{X = k X A}if: (a) A = {X > m}wheremisapositiveinteger. (b) A = {X < m}. (c) A = {X isanevennumber}. Commentontheshapeoftheconditionalpmfofpart(a)... Conditional cdf. Let A be a nonzero probabilit event A. Show that (a) P(A) = P(A X )F X () + P(A X > )( F X ()). P(A X ) (b)f X ( A) = F P(A) X ()... Joint cdf or not. Consider the function G(, ) = if + 0, 0 otherwise. CanG beajointcdfforapairofrandomvariables? Justifouranswer... Time until the n-th arrival. Let the random variable N(t) be the number of packets arriving during time(0, t]. Suppose N(t) is Poisson with pmf p N (n) = (λt)n e λt forn = 0,,2,... n! LettherandomvariableY bethetimetogetthen-thpacket. FindthepdfofY.

18 8 Pairs of Random Variables.. Diamond distribution. Consider the random variables X and Y with the joint pdf where c isaconstant. (a) Find c. (b)find f X ()and f X Y ( ). f X,Y (, ) = c, if + / 2, 0, otherwise, (c) Are X and Y independent random variables? Justif our answer. (d)definetherandomvariable Z = ( X + Y ). Findthepdf f Z (z)... Coin with random bias. You are given acoin but are nottold whatits bias (probabilit of heads) is. You are told instead that the bias is the outcome of a random variable P U[0,]. To get more information about the coin bias, ou flip it independentl times. Let X be the number of heads ou get. Thus X Binom(0,P). Assuming that X = 9, find and sketch the a posteriori probabilit ofp,i.e., f P X (p 9)... First available teller. Consider a bank with two tellers. The service times for the tellers are independenteponentiall distributed randomvariables X Ep(λ ) and X 2 Ep(λ 2 ), respectivel. You arrive at the bank and find that both tellers are bus but that nobod else is waiting to be served. You are served b the first available teller once he/she is free. (a) Whatistheprobabilit thatouareservedbthefirstteller? (b)lettherandomvariabley denoteourwaitingtime. FindthepdfofY... Optical communication channel. Let the signal input to an optical channel be given b X = withprobabilit 2 0 withprobabilit 2. TheconditionalpmfoftheoutputofthechannelY {X = } Poisson(),i.e.,Poissonwithintensit λ =,andy {X = 0} Poisson(0). (a) ShowthattheMAPrulereducesto D() =, < 0, otherwise. (b)find andthecorrespondingprobabilit oferror... Iocane or Sennari. An absent-minded chemistr professor forgets to label two identicall looking bottles. One bottle contains a chemical named Iocane and the other bottle contains a chemical named Sennari. It is well known that the radioactivit level of Iocane has the U[0, ] distribution, while the radioactivit level of Sennari has the Ep() distribution.

19 Problems 9 (a) Let X be the radioactivit level measured from one of the bottles. What is the optimal decision rule (based on the measurement X) that maimizes the chance of correctl identifing the content of the bottle? (b) What is the associated probabilit of error?.. Independence. Let X X and Y Y be two independent discrete random variables. (a) Showthatantwoevents A X andb Y areindependent. (b)showthatantwofunctionsof X andy separatel areindependent;thatis,if U = д(x) andv = h(y)thenu andv areindependent... Familplanning.AliceandBobchooseanumberX atrandomfromtheset{2,3,4} (so the outcomes are equall probable). If the outcome is X =, the decide to have children until the have a girl or children, whichever comes first. Assume that each child is a girl with probabilit / (independent of the number of children andgenderofotherchildren). LetY bethenumberofchildrenthewillhave. (a) Findtheconditionalpmf p Y X ( )forall possiblevaluesof and. (b)findthepmfofy... Radarsignaldetection. ThesignalforaradarchannelS = 0ifthereisnotargetand arandomvariables N(0,P)ifthereisatarget. Bothoccurwithequalprobabilit. Thus S = 0, withprobabilit 2 X N(0,P), withprobabilit 2. The radar receiver observesy = S + Z, wherethenoise Z N(0,N) is independent of S. Find theoptimal decoder for deciding whethers = 0 or S = X and its probabilit of error? Provide our answer in terms of intervals of and provide theboundarpointsoftheintervalsintermsofp andn... Ternar signaling. Let the signal S be a random variable defined as follows: withprobabilit 3 S = 0 withprobabilit 3 + withprobabilit. 3 ThesignalissentoverachannelwithadditiveLaplaciannoiseZ,i.e., Z isalaplacian random variable with pdf f Z (z) = λ 2 e λ z, < z <. ThesignalSandthenoiseZareassumedtobeindependentandthechanneloutput istheirsumy = S + Z.

20 20 Pairs of Random Variables (a) Find f Y S ( s) for s =, 0, +. Sketch the conditional pdfs on the same graph. (b)find the optimal decoding rule D(Y) for deciding whether S is, 0 or +. Give ouranswerintermsofrangesofvaluesofy. (c) Findtheprobabilit ofdecodingerrorford()intermsof λ... Signal or no signal. Consider a communication sstem that is operated onl from time to time. When the communication sstem is in the normal mode(denoted b M = ),ittransmitsarandomsignals = X with +, with probabilit /2, X =, with probabilit /2. When thesstem is in the idle mode (denoted b M = 0), it does not transmit ansignal(s = 0). Bothnormalandidlemodesoccurwithequalprobabilit. Thus X, withprobabilit /2, S = 0, with probabilit /2. The receiver observes Y = S + Z, where the ambient noise Z U[,] is independent of S. (a) Find and sketch the conditional pdf f Y M ( ) of the receiver observation Y giventhatthesstemisinthenormalmode. (b)find and sketch the conditional pdf f Y M ( 0) of the receiver observation Y giventhatthesstemisintheidlemode. (c) Findtheoptimaldecoder d ()fordecidingwhetherthesstemisnormalor idle. Provide theanswerintermsofintervalsof. (d) Find the associated probabilit of error... Two independent uniform random variables. Let X and Y be independentl and uniforml drawn from the interval[0, ]. (a) FindthepdfofU = ma(x,y). (b)findthepdfofv = min(x,y). (c) FindthepdfofW =U V. (d)findthepdfof Z = (X +Y) mod. (e) Findtheprobabilit P{ X Y /2}... Two independent Gaussian random variables. Let X and Y be independent Gaussian random variables, both with zero mean and unit variance. Find the pdf of X Y.

21 Problems 2.. Functions of eponential random variables. Let X and Y be independent eponentiall distributed random variables with the same parameter λ. Define the followingthreefunctionsof X andy: U = ma(x,y), V = min(x,y), W =U V. (a) FindthejointpdfofU andv. (b)findthejointpdfofv andw. Aretheindependent? Hint:Youcansolvepart(b)eitherdirectlbfindingthejointcdforbepressing thejointpdfintermsof f U,V (u, )andusingtheresultofpart(a)... Maimalcorrelation.Let(X,Y) F X,Y (, )beapairofrandomvariables. (a) Show that F X,Y (, ) min{f X (),F Y ()}. NowletF()andG()becontinuous andinvertible cdfsandlet X F(). (b)findthecdfof Y = G (F(X)). (c) Show that F X,Y (, ) = min{f(), G()}.