probabilism PDF affltldt Note The numerical value f f should rest be viewed as a Plaekeb Pftexettdt PK t o htt c IR PCXeB fltldt II Reading 4 l t 3

Transcription

1 Reading 4 l t 3 Densities and Probabilities We now consider rado variables that can take an uncountable set called continuous RVs values in A RV X is continuous if PCXeB fltldt II It fhdt where Lyft is called the probabilism PDF We're often interested in sets 13 bury the form E b typically consider tasks such as computing Plaekeb affltldt so we Note The numerical value f f should rest be viewed as a probability For continuous RVs PK t o htt c IR Instead think of Heldt as the element of probability Pftexettdt

2 Common Densities uniform out radon b t lies in some f ft a t't otherwise known interval 1 y exponential non negative RVs that decay over the a us f I t Laplace like a double sided exponential 14 Ee HH 1 o Gaussian aka normal most commonlimp tat densify arises in any applications and rice to work with f ft expf HELI a E

3 We will see in a few beefs that if we Lace RVs Y Xz Xn all independent and standardized they Xi Ncaa as n sa called the central kit theorem Fact No function exists from 5 End to the power at 2 such that D P Ecb b a to Easbel 2 the axioms of probability are satisfied p Search Vitali set beyond scope of course If the joint density fxylx.gl exists we say X and Y are independent if fxy x g 611 g Epeckk Expectation is analogous to the discrete case ELgktJ fgcxlfxlxld.fr ay Latin g R R I particular EE Ix f f dx

4 Transfor Methods 1 A few transformations make Computing functions of RVs ties so The m watyhf.fr GF of a RV X 7 Mels Elle IE fxhdx easier As the name indicates the MOF can be used to compute mounts i e ELI'T of a RU X It holds that Xk 2k Mx b too In words the Kth moment of X c be found by taking the Kth devotee of the NGF ad evaluating at s o EJ xnnco.dk rt EEesxJ fe e eh42 I e e dx dx Nfs o PDF

5 New note 2 texts set FLEX y M s s et te checks EEXY.lt at The MGE shoes up when applying Chernoff's bandy method as well which we'll see in a few weeks Dgp If X Y are independnt RVs and Z XtY then Proof Mz s Mx1st My1st Mzls e Zfz z de x fesktpfxylg.xldxdg.ie by independence e etjfxlxlf.ly d dgotx.y fe fxhdr.fetfylgldg MxlslMyls

6 Robley a Let X be a non negative RV Show tht E Tx f Past dt b Let the CDF of be Ix PIX ex l e 2 so Ed FLEX Scottie twist p ingrate Isaak IPlxsttdt e.iefxhdxdtf IIfxHdtdx flute horizontal strips Hxdx EET b EEx f PIX't dt l Elt It e Edt II

7 DetectionTheoyriaut comunications ad machine learning we are interested in classifying signals measurements in a statistically optial way Tr suit signal X through a bang channel receive signal Y Gwen received signal what should we say was transmitted Exf Feature label pairs X y generated according to Seo distributee Rey Gwen a new feature vector decide as class Tf We can use probability theory to optimize our decisions For now ss X c o I biog Setup classification communications a Ho null hypothesis 4 0 Do decide He H alternate hypothesis X i D i decide H error occurs if false alarm Tfesfeftion c Ho and decide X I Horn 2 tf and decide X o o Hinde Goal Design detector to minimize probability of error

8 PCE P1 Hora UCH ND P HorD DCD.lt o PCH noo PCHoJtPCD 1HDPCH We make our decision by separating the space Ro Rc regions e to two ff lho y errors come f Ia'Isese fifth Ro decide Ho threshold Re decide H Minimizing the probability of error the amounts to the minimizing probability cross from each Ho and H that lies on the wrong side of the boundary We then want to minimize PE 1 PITH filth dx t f Pl Ho fhho dx Note that I IHH dx p flxltlddxts.lk Hddr

9 So we can rewrite PCE as PIE RHO SHIH dx PCH PCH fcxlthdx R Split fg1h PIH.IS xlhi dx PCH 121 Note that we can't change PCH so we should choose Rs to minimize R PCH AHH PCH.lt xlhi dr which is done by making R the region over which PCH fcx H This leads to the PCH flxlho decide Ho MAP detection rule t H 11 7 If we divide both sides by f fx the relation to the posterior probability is more obvious the r le g.mg t TIE I H

10 If we do not have priors on the two events Ha Ha we can assume they are equal yielding detector the maximu I f Htt Z f IHH ML Rule likelihoody ML Reminder PCH is the prior probability that Ho before happens any data is received is the probability that Ho happens now that posterior PC Holy we have the data X PIX1H is the that liked that He took place Note the MAP rule is also equi f filth H plhol flxlho PCH takes some value goes leet to We call this the liheliheodr.at where equalpriors make the RHS equal to 1 and the Ma detector gie

11 Problemy Assume we transmit a signal at one volts which is then corrupted by additive NCO T noise Compute the likelihood ratio Solution Fron the description we have H Y MTN Ho Y N where N N o T2 We then have SHIH 7 expf SHIH e.pl and the ratio is f Ith to therefore expf m2 2ym