Exploring Basic Probability V. Krishnan Presentation to CACT - September 28, 2001

Transcription

1 Eplorig Basic Probability V. Krisha Presetatio to CACT - September 8, Abstract I this talk we will discuss some basic ideas o Probability like idepedece, coditioal probability, Bayes' theorem, aspects of combiatorics, hypergeometric distributio, upper ad lower bouds o the Gaussia tails, cocludig with some cotiuous distributios. A few of the results are ew.. Coditioal Probability ad Idepedece The coditioal probability of a evet B coditioed o the occurrece of aother evet A is defied by, } P{B A} P{A B} P{A} if P{A} (.) P{A B} P{B A} P{A} If P{A} the the coditioal probability P{B A} is ot defied. Sice this is a probability measure it also satisfies Kolmogorov aioms. Idepedece: Two sets A ad B ca be called fuctioally idepedet if the occurrece of B does ot i ay way ifluece the occurrece of A ad vice versa. Fuctioal idepedece is a differet cocept from statistical idepedece that will be defied later. For eample, the tossig of a coi is fuctioally idepedet of the tossig of a die because they do ot deped o each other. However, the tossig of a coi ad die are ot mutually eclusive sice ay oe ca be tossed irrespective of the other. By the same toke, pressure ad temperature are ot fuctioally idepedet because the physics of the problem, Boyle s law coects these quatities. They are certaily ot mutually eclusive. We will ow defie statistical idepedece of evets. If A ad B are two evets i a field of evets, the A ad B are statistically idepedet if ad oly if, P{A B} P{A} P{B} (.) Statistical idepedece either implies or implied by fuctioal idepedece. Uless otherwise idicated we will call statistically idepedet evets as idepedet without ay qualifiers. Eample. (Statistically idepedet but fuctioally depedet) Two dice, oe red ad the other blue are tossed. These tosses are fuctioally idepedet ad we have the cartesia product of elemetary evets i the combied sample space with each evet beig equiprobable. We seek the probability of a evet B defied by the sum of the umbers showig o the dice equals 9. There are four poits {(6, ), (5, ), (, 5), (, 6)} ad hece P{B} 6 9. We ow coditio the evet B with a evet A defied as the red die shows odd umbers. The probability of the evet A is P{A} 8 6. We wat to determie whether the evets A ad B are statistically idepedet. These evets are show i Fig.(.) Fig..

2 From Fig.(.) P{A B} P{( 6), (5 )} 6 8. ad P{A} P{B} 9 8 showig statistical idepedece. We compute P{B A} from the reduced sample space poit of view. The coditioig evet reduces the sample space from 6 poits to 8 equiprobable poits ad the evet {B A} {( 6), (5 )}. Hece P{B A} 8 9 P{B}, or, the coditioig evet has o ifluece o B. Here, eve though the evets A ad B are fuctioally depedet, they are statistically idepedet. Eample. (Statistically depedet ad fuctioally depedet) However, if aother set C is defied by the sum beig equal to 8 as show i Fig.(.) the P{C} Fig.. Here the evets C ad A are ot statistically idepedet because P{C}.P{A} P{C A} 7. I this eample, we have the case where the evets A ad C are either statistically idepedet or fuctioally idepedet. Eample. (Statistically depedet but fuctioally idepedet) We shall ow take a eample where the evets are fuctioally idepedet but statistically depedet. A ur cotais 5 balls out of which are red, 5 are gree ad 6 are blue. Let X be the radom variable represetig the red balls, Y the radom variable represetig the gree balls ad Z the radom variable represetig the blue balls. Sice X + Y + Z 5 ay two of the three radom variables are fuctioally idepedet i the sese that they ca be chose idepedetly of each other. We shall assume that X ad Y are fuctioally idepedet of each other while Z 5 X Y. We pick balls out of this ur ad calculate the joit probability P XY (, y) of drawig red ad gree balls. This probability is give by, ( 5 ) 6 P XY (, y) ( y ) ( ) We ca ow obtai the margial probabilities P X () ad P Y (y) as y P X () P XY (, y) y y P Y (y) P XY (, y) y +y 5 ( 5 ) ( ) ( 6 ) y ( ) 5 9 y y 5 The probabilities P Y X (y X), Y coditioed o X, ad P X Y ( Y), X coditioed o Y ca be give as, P Y X (y X) P XY (, y) P X () ( 6 ) y y y,,,,

3 P X Y ( Y) P XY (, y) P Y (y) ( 5 ) y 9 y y y,,,, Sice P Y X (y X) is ot equal to P Y (y) ad P X Y ( Y) is ot equal to P X () we coclude that X ad Y are ot statistically idepedet eve though thet are fuctioally idepedet. Coditioal Probability ad Reduced sample space Coditioal probability P{B A} ca also be iterpreted as the probability of the evet B i the reduced sample space give by the set differece (S B). Sometimes it is simpler to use the reduced sample space to solve coditioal probability problems. We shall use both these iterpretatios i obtaiig coditioal probabilities as show i the followig eamples. Eample. The game of craps as played i Las Vegas has the followig rules. A player rolls two dice. He wis o the first roll if he throws a 7 or a. He loses if the first throw is a, or. If the first throw is a, 5, 6, 8, 9 or it is called a poit ad the game cotiues. He goes o rollig util he throws the poit for a wi or a 7 for a loss. We have to fid the probability of the player wiig. We will solve this problem both from the defiitio of coditioal probability ad the reduced sample space. The followig table shows the umber of ways the sums,,, 5, 6, 7, 8, 9,,, of the umbers appearig o the two dice ca appear ad their probabilities. Die A 5 6 Die B Total P{Total} Solutio usig defiitio of coditioal probability 6 The probability of wiig i the first throw is P{7} + P{} The probability of 6 losig i the first throw is P{} + P{} + P{} To calculate the probability of wiig i the secod throw we shall assume that i is the poit with probability p. The probability of ot wiig i ay give throw is give by the probability of ot i ad ot 7 r p. We compute the coditioal probability 6 P{wi i i first throw} p + rp + r p + a ifiite geometric series. This sum is give by, p P{wi i i first throw} r, ad p.p P{wi i secod or subsequet throws} r The probabilities of makig the poit, 5, 6, 8, 9, is obtaied from the table give above. Thus for i,

4 P{wi after first with i } P{wi after first with i 5} P{wi after first with i 6} ad the same probabilities for 8, 9,. Thus the probability of wiig i craps is give by: P{wi} P{wi i roll } + P{wi i roll,, } 8 { 6 } +. { } Solutio usig Reduced Sample Space We ca ow use the reduced sample space to arrive at the same result i a much simpler maer. After the poit i has bee rolled i the first throw the reduced sample space cosists of (i +6) poits for i, 5, 6 ad ( i+6) poits for i 8, 9, i which the game will termiate. We ca see from the table that there are 6 ways of rollig 7 for a loss, (i, i, 5, 6) ways of rollig the poit for a wi ad ( i, i 8, 9, ) ways of rollig the poit for a wi. Thus, P{wi i roll,, i i roll } P{W i} i i, 5, 6 i +6 i i+6 i 8, 9, Thus, P{W }, P{W 5}, P{W 6} 5, P{W 8} 5, P{W 9} 9, P{W } 9. Hece probability of a wi after the first roll is P{W i}.p{i}. Hece, P{wi after first roll} P{wi} P{wi i first roll} + P{wi after first roll} A result obtaied with comparative ease.. Bayes' Theorem The et logical questio to ask is, "Give that the evet B has occurred what are the probabilities that A, A, A are ivolved i it?" I other words, we have to fid the coditioal probabilities P(A B}, P(A B},, P(A B}. Bayes theorem ca be writte as, P{A k B} P{A kb} P{B} P{B A k}p{a k } P{B} P{B A k }P{A k } i P{B A i }P{A i } This theorem coects the a priori probability P{A i } to the a posteriori probability P{A i B}. This useful theorem has wide applicability i commuicatios ad medical imagig. Eample. Moty Hall Problem This problem is called the Moty Hall problem because of the game show host Moty Hall who desiged this game. There are doors A, B ad C, ad behid oe of them is a car ad behid the others are goats. The cotestat is asked to select ay door ad the host opes oe of the other two doors ad shows a goat. He (.)

5 the offers the choice to the cotestat of switchig to the other uopeed door or keep the origial door. The questio ow is whether the probability of wiig the car is improved if he switches or it is immaterial whether he switches or ot. The aswer is couter ituitive i the sese that oe maybe misled ito thikig that it is immaterial whether oe switches or ot. Mathematical Aalysis Let us aalyze this problem from the poit of view of Bayes theorem. We shall first assume that the host kows the door behid which the car is located. Let us also assume that the host opes door B give that the cotestat s choice is door A. The apriori probabilities of a car behid the doors A, B ad C are, P{A} : P{B} : P{C} We ca make the followig observatios. If the choice of the cotestat is door A ad the car is behid door A the the host will ope the door B with probability of (sice he has a choice betwee B ad C). O the other had, if the car is behid door B, the there is zero probability of his opeig door B. If the car is behid door C, the he will ope door B with probability. Thus we ca write the followig coditioal probabilities. P{B A} : P{B B} : P{B C} We ca ow calculate the total probability P(B) of the host opeig door B. P{B} P{B A}P{A}+P{B B}P{B}+P{B C}P{C} Usig Bayes theorem eq. (.) we ca ow fid the aposteriori probabilities of the car behid the doors A or C (B has already bee opeed by the host) coditioed o B. These are: P{A B} P{BA} P{B} P{C B} P{BC} P{B} P{B A}P{A} P{B} P{B C}P{C} P{B}.. Similar aalysis holds for other cases of opeig the doors A or C. We ca see from the above result that the cotestat must switch if he wats to double his probability of wiig the car.. Combiatorics May problems i combiatorics ca be modeled as placig r balls ito cells. Permutatios The two ways of cosiderig permutatios are (i) samplig with replacemet ad with orderig (ii) samplig without replacemet ad with orderig Combiatios Similar to the case of perturbatios we have two ways of cosiderig combiatios, (i) samplig without replacemet ad without orderig (ii) samplig with replacemet ad without orderig. We will oly cosider samplig with replacemet ad without orderig. Samplig with replacemet ad without orderig We shall ow cosider formig r-combiatios with replacemet ad without orderig. I this case we pick r balls with replacemet from balls i distict categories like, red, blue, gree brow. We do ot distiguish amog balls of the same color ad the distictio is amog differet colors. For eample, if we are pickig 6 balls from the color categories the it ca be arraged as follows.

6 Red Blue Gree Brow * * * * * * There are ow red balls, blue ball, o gree balls ad brow balls. The bar symbol separates the four categories. The arragemet ca ot start with a bar or ed with a bar. Hece we have bars. We ow have to choose 6 balls from 6 balls plus bars without replacemet ad without orderig. There are the ways of pickig 6 balls from differet categories. 6 Geeralizig this result, if there are distict categories of balls ad if we have to pick r balls without regard to order ad with replacemet, the Number of r-combiatios + r (.) r This is the celebrated Bose-Eistei statistics where the elemetary particles i the same eergy state are ot distiguishable. Eample. Here is a very iterestig eample to illustrate the versatility of the formula i eq. (.). dice each with faces marked,,, are tossed. We have to calculate the umber of ways the tosses, will result i eactly umber, umbers, umbers ad umbers showig o the faces of the dice without regard to order ad with replacemet. The total umber of ways is tabulated below i Tab.(.) Table. sigle digits 8 double digits triple digits quadruple digit We ca see from the above eample that there are oe-digits, 8 two-digits, three-digits ad four-digits totalig 5 arragemets without regard to order ad with replacemet. The problem ca be stated as follows. Four -face dice are tossed without regard to order ad with replacemet. Fid the umber of ways, a. this ca be achieved. With ad r i eq. (.), the umber of ways is that agrees with the total umber i the table. b. eactly oe digit is preset. The umber of ways digit ca be draw from digits is ( ) ( ) ways. The umber of ways ay particular digit for eample {} will occupy a digit umber is a restricted case with occupyig oe slot ad the rest of the three are urestricted. With ad r i eq. (.), the umber of ways for this particular sequece {} is + ( ) ( ) ( ). The total umber usig all the combiatios are. ( ) ( ) ways. These ways are (), (), (), (). c. eactly digits are preset. The umber of ways digits ca be draw from digits is ( ) 6 ways. The umber of ways ay particular digits, for eample, {, } will occupy a digit umber is a restricted case with both ad occupyig ay two slots ad the other the two are urestricted i the sese they ca be occupied by

7 either, or (, ). With ad r i eq. (.), the umber of ways for this particular sequece {, } is + ( ) ( ) ( ). The total umber usig all the ( ) combiatios are ( ) ( ) 8. ways. This agrees with the eumeratio. d. eactly digits are preset. The umber of ways digits ca be draw from digits is ( ) ways. The umber of ways digits ca be arraged i slots such that all the three digits are preset is agai give by the restrictio formula + ( ) ( ) ( ) ways. Hece the total umber of ways is ( ) ( ). ways agreeig with the eumeratio e. eactly digits preset. The umber of ways digits will be preset is. This ca be obtaied from the argumet i (c) above givig, ( ) + ( ) ( ) ( ) ( ) way. Geeralizatio The above result ca be geeralized to the tossig of -sided dice. The total umber of sample poits without regard to order ad with replacemet is from eq. (.) + (.) Out of these the umber of ways eactly k digits will occur ca be derived as follows: The umber of ways k digits ca be chose from is ad if ay particular sequece of k digits has to fill k slots the the umber of ways this ca be doe is obtaied by substitutig k ad r ( k) i eq. (.) yieldig k + ( k) k. (.) k Sice all the ( k ) sequeces are fuctioally idepedet, the umber of ways N(, k) is from eq. (.), N(, k) k k (.) Usig eq. (.) we have tabulated N(, k) for,, 8 ad k,, 8 i Tab.(.) T -sided die throw times Table..

8 I additio, we have the followig iterestig idetity. Combiig eqs.(.) ad (.) we have, k k k (.5). Hypergeometric Distributio The biomial distributio is obtaied while samplig with replacemet. Let us select a sample of microprocessor chips from a bi of N chips. It is kow that K of these chips are defective. If we sample with replacemet the the probability of ay defective chip is K N. The probability that k of them will be defective i a sample size of is give by the biomial distributio sice each drawig of chips costitute Beroulli trials. O the other had, if we sample without replacemet, the the probability for the first chip is K N, the probability for the secod chip is K N ad for each succeedig drawig the probabilities chage. Hece the drawigs of the chips do ot costitute Beroulli trials ad biomial distributio does ot hold. Hece we have to use alterate methods to arrive at the probability of fidig k defective chips i trials. The umber of sample poits i this sample space is choosig chips out of the total N chips without regard to order. This ca be doe i ( N ) ways. Out of this sample size of, there are k defective chips ad ( k) good chips. The umber of ways of pickig k defective chips out of the total of K defective chips is ( ) K k ad the umber of ways of drawig ( k) good chips out of the remaiig (N K) good chips is N K. We k deote the total umber of ways of drawig a sample of chips with k bad chips as the evet E. Sice these pickigs are fuctioally idepedet the umber of ways is ( K k ) N K. Hece the probability of the k evet E is give by, K N K k k P{E k} This is kow as the Hypergeometric Distributio. N Eample. Hypergeometric distributio is used to estimate a ukow populatio from the data obtaied. To estimate the populatio N of tigers i a wildlife sactuary, K tigers are caught, tagged ad released. After the lapse of a few moths, a ew batch of tigers is caught ad k of them are foud to be tagged. It is assumed that the populatio of tigers does ot chage betwee the two catches. The total umber of tigers N i the sactuary is to be estimated. I eq. (.), the kow quatities are K, k ad. The total umber N has to be estimated i equatio K N K P k (N) k k N We will fid the value of N that maimizes the probability P k {N}. Such a estimate is called the maimum likelihood estimate. Sice the umbers are very large we ivestigate the ratio: (.) P k (N) P k (N ) K k N K k N K k N K k N

9 P This ratio k (N) is greater tha if, P k (N ) N K k N N N N K N K-+k (N ) (N K) > N(N K +k) or, Nk < K, or, N < K k. N N K k Coclusio: If N is less tha K k the P k(n) is icreasig ad if N is greater tha K k the P k(n) is decreasig. Hece the maimum value of P k (N) occurs whe N is equal to the earest iteger ot eceedig K k 5 ad k the the populatio of tigers is.5 5 tigers.. Thus if K, 5. Upper ad Lower Bouds for Gaussia Tails Q() As metioed i property (7), eq. (6..8), Q() is used to epress the bit error probability i trasmissio of commuicatio sigals. Sice Q() does ot have a closed form solutio, bouds ca be established. Q() ca epaded ito a asymptotic series give by, Q() e π { } ( ).. ( ) + (5.) ad for large, Q() ca be approimated by, Q() e π (5.) This is ot a very good boud for small. A very tight boud has bee established i Ref[?] by first itegratig Q() by parts as show below: Q() π e ξ dξ π e π ξ ξ e ξ dξ e ξ dξ ξ? ad the epressio for Q() ca be maipulated to obtai, Q() ( α) + α +β π e (5.) α ad β are optimized by miimizig the absolute relative error fuctio ε() give by, ε() Q() Q () Q() The upper boud QU() is obtaied for values of α. ad β 5.5 ad the lower boud QL() is obtaied for values of α / π ad β π. Thus,

10 ad QU() QL() ( π ) + π +π π e π e The fuctios Q(), QU() ad QL() are show i Fig. (5.) ad tabulated i Tab.(5.). This table shows that the upper ad lower bouds are for all >. The percetage errors are also tabulated i Tab.(6.) ad show i Fig.(5.) Fig. 5. Lower Boud True Value Upper Boud Table %Error o Lower Boud %Error o Upper Boud Table 5.

11 .5 Percet Error.5 Percet Error o the lower boud.5 Percet Error o the upper boud 5 Fig. 5. The figures ad tables illustrate clearly that the approimatio for all is ecellet with the maimum error of about % occurrig at low values of. 6. Some Cotiuous Distributios Chi-Square Distributio The Laplace, Erlag, Gamma ad Weibull distributios are based o epoetial distributios. The Chi - square distributio straddles both the epoetial ad Gaussia distributios. The geeral gamma distributio has parameters (α, λ). I the gamma distributio if we substitute α / ad λ / we obtai the Chi - square distributio defied by, e ( ) > f X () Γ ( ) (6.) where is called the degrees of freedom for the χ -distributio. We ca itroduce aother parameter σ ad rewrite eq. (7.7.) as e ( ) σ > f X () ( σ ) Γ ( ) (6.) If we substitute χ i eq. (6.) we obtai the familiar form of the Chi-square distributio give by, ( χ ) e ( χ σ ) > f X () ( σ ) Γ ( ) (6.) where Γ(α) is the gamma fuctio defied by the itegral, Γ(α) α e d, α > (6.) Itegratig eq. (6. by parts with u α ad dv e d, we have, Γ(α) α e d, α > a e + (α (α )Γ(α ) ) α e d If α is a iteger, the Γ() ( )Γ( ) ad cotiued epasio yields Γ()!. Hece the gamma fuctio ca be regarded as the geeralizatio of the factorial fuctio for all positive real umbers. The most useful fractioal argumet for the gamma fuctio is / with, Γ π (6.6) The icomplete gamma fuctio Γ (α) is defied by, (6.5)

12 Γ (α) ξ α e ξ dξ has bee tabulated. We will use eq. (6.7) later for the χ -distributio fuctio F X (). (6.7) If is eve i eq. (6.) the Γ! ad if is odd the Γ i eqs. (6.) (6.) is obtaied by iteratig Γ with the fial iterat Γ π. Eq. (6.) ca also be obtaied from the followig: If Y, Y,, Y are i.i.d Gaussia radom variables with zero mea ad variace σ, the X X k is χ - distributed with -degrees of freedom. k The cumulative distributio fuctio F X () is give by itegratig eq. (6.), F X () ξ e ( ξ ) σ dξ ( σ ) Γ ( ) > (6.8) Eq. (6.8) ca be give i terms of the icomplete gamma fuctio Γ X (α) give by eq. (6.7). Substitutig η ξ/σ ad dξ σ dη i eq. (6.8) we obtai, F X () ξ e ( ξ ) σ ( σ ) Γ ( ) σ dξ (6.9) σ η e ( η ) dη Γ ( σ ) Γ ( ) Γ ( ) Whe the Γ σ Γ(/) ad hece F X (). The cdf F X () i eq. (6.9) has bee etesively tabulated. The Chi-square distributio is show i Fig. (6.) for the umber of degrees of freedom,,, Fig. 6. The Chi-square distributio is used directly or idirectly i may tests of hypotheses. The most commo use of the chi-square distributio is to test the goodess of a hypothesized fit. The goodess of fit test is performed to determie if a observed value of a statistic differs eough from a hypothesized value of a distributio to draw the iferece whether the hypothesized quatity is ot the true distributio. Eample 7. The desity fuctio of f Z (z) of Z X + Y where X ad Y are idepedet radom variables distributed as N(, σ ) ca be foud by substitutig i eq. (6.) yieldig,

13 e ( z ) σ z > (z) σ z f Z which is a χ -desity with degrees of freedom. The correspodig cdf, F Z (z) is give by, F Z (z) e { ( z ) σ z > z Chi-Distributio The Rayleigh ad Mawell distributios are special cases of the Chi distributio. Just as the sum of the squares of zero mea Gaussia radom variables with variace σ is Chi-square distributed with degrees of freedom we ca cosider that the sum of the square root of the sum of the squares of zero mea Gaussia radom variables with variace σ is Chi distributed with degrees of freedom. Accordigly, let Z be the radom variable give by, Z X + X + + X (6.) where {X i. i,, } are zero mea Gaussia radom variables with variace σ. The the chi-desity f Z (z) ca be obtaied by substitutig z ad d zdz i the χ -desity give by, e ( ) σ f X () d d ( σ ) fz (z) dz Γ ( ) ad f Z (z) is evaluated to yield, f Z (z) Γ ( ) where Γ(/) is the gamma fuctio defied i eq. (6.). z e ( z σ ) z > σ z (6.) A family of Chi-distributios is show i Fig. (6.) for σ.5 ad,,, 5, 7, 9,,, 5..6 f X () σ.5 Rayleigh Mawell Fig. 6. The distributio fuctio F Z (z) is give by itegratig eq. (6.), F Z (z) Γ ( ) z ζ σ e ( ζ σ ) dζ z > z (6.)

14 There is o closed form solutio for eq. (7.8.) but it ca be epressed i terms of the icomplete gamma fuctio Γ z (z) defied i eq. (6.7). By substitutig y ζ σ ad dζ (σ dy) ζ i the i eq. (6.) we ca write, F Z (z) Γ ( ) Γ ( ) z σ y e y dy Γ( z σ ) ( ) z σ ( ) Γ ( ) Γ( ) (6.) where Γ z σ is the icomplete gamma fuctio defied i eq. (6.7). Clearly as z, Γ z σ Γ(/) thus showig that F Z ( ). There are two very importat special cases of the Chi-distributio whe ad. Rayleigh Distributio Whe i the Chi distributio of eq. (7.8.) we have, f Z (z) { } z e ( ) σ z σ z > z (6.) This is called the Rayleigh desity ad fids wide applicatio i commuicatios fadig chaels. If we have two idepedet zero mea Gaussia radom variables X ad Y with variace σ, the if we make the trasformatio X Z cosθ ad Y Z siθ, the Z Z X + Y is Rayleigh distributed which is a chidistributio with degrees of freedom. θ is uiformly distributed i the iterval (, π]. The Rayleigh desity is show i Fig. (6.) for σ,, 5.. σ.5 σ σ Fig. 6. The cdf correspodig to the Rayleigh desity ca be epressed i a closed form as follows: F Z (z) z ζ σ e ζ σ dζ Or, F Z (z) { e z σ z > z (6.5) Mawell Desity This is oe of the earliest distributios derived by Mawell ad Boltzma. They derived that the speed distributio of gas molecules is give by, f(v) π m π kt v e mv kt (6.6)

15 where k is the Boltzma costat, m is the mass, T is the absolute temperature ad v is the speed. This is based o the assumptio that the molecules are distiguishable ad they ca occupy ay eergy state. This is a special case of the Chi-desity of eq. (7.8.) with degrees of freedom ad is give by, f X () e σ (6.7) π σ If we substitute kt m σ i the Mawell-Boltzma distributio of eq. (6.6), we obtai eq. (6.7). The pdf f X () for the Mawell ad Rayleigh desities are show i Fig. (6.) for values of σ, ad Solid Lie Mawell Dotted Lie Rayleigh σ σ σ Fig. 6. The cdf F X () correspodig to the Mawell desity f X () is obtaied by itegratig eq. (6.7) ad is give by, F X () π ξ e ξ σ σ dξ (6.8) There is o closed form solutio for eq. (7.8.9). However, itegratig by parts, it cam be epressed as, F X () π σ e ξ σ dξ π σ e σ erf σ π σ e σ > (6.9) where the error fuctio erf() is defied as, erf() π e ξ dξ (6.) Rice s Desity I the Rayleigh distributio discussed i the last sectio, the radom variables X ad Y must be zero mea havig the same variace σ. I may commuicatio problems ivolvig fadig chaels, X ad Y will ot be zero mea ad will have mea values equal to µ X ad µ Y. If the radom variable Z X + Y the f Z (z) has a Rice desity with a ocetrality parameter m µ X + µ Y. It is give by, f Z (z) z ep (z + m I mz (6.) σ σ σ where I (z) is the zeroth order modified Bessel fuctio of the first kid. The geeral νth order modified Bessel fuctio is give by the series, z k I ν (z) ( z ) ν ν ( ) k k! Γ(ν + k + ) (6.)

16 ad the zeroth order is give by, ( ) I (z) k ( k! ) z k (6.) ad I (), I (), I (), I () are show i Fig. (6.5). 5 I () I () I () I () Fig. 6.5 From Fig. (6.5) we see that I () ad hece whe m (zero mea) the f X () is the Rayleigh distributed. Rice s Desity is show i Fig. (6.6) for m,,, 6, 8,,,. m is the Rayleigh desity.. m: Rayleigh σ.5 m m m6 m8 m m m Fig. 6.6 The distributio fuctio F X () correspodig to eq. (6.) is give by itegratig f Z (). F Z (z) z ζ ep (ζ + m I σ σ mζ dζ σ (6.) There is o closed from solutio for this itegral. Nakagami Desity The Nakagami distributio, like the Rayleigh desity also belogs to the class of cetral Chi-square distributios. It is used for modelig data from multipath fadig chaels ad it has bee show to fit empirical results more geerally tha the Rayleigh distributios. Sometimes, the pdf of the amplitude of a mobile sigal ca also be described by the Nakagami m-distributio. The Nakagami distributio may be a reasoable meas to characterize the backscattered echo from breast tissues to automate a scheme for separatig beig ad maligat breast masses. The shape parameter m has bee show to be useful i tissue characterizatio. Chi-square tests showed that the this distributio is a better fit to the evelope tha the Rayleigh distributio. Two parameters, m (effective umber) ad Ω (effective cross sectio), associated with the Nakagami distributio are used for the effective classificatio of breast masses.

17 The Nakagami m-desity is defied by, f X () Γ(m) m Ω m m e m Ω (6.5) where m is the shape parameter ad Ω cotrols the spread of the distributio. The iterestig feature of this desity is that ulike the Rice distributio, it is ot depedet o the modified Bessel fuctio. The Nakagami family is show i Fig. (6.7) for Ω ad for m,,.5,,.75, Ω m m m /6 m m /6 m / Fig. 6.7 The cdf F X () is give by, F X () Γ(m) m Ω m ξ m e m ξ Ω dξ (6.6) This has o closed form solutio. However, it cam be give i terms of the icomplete gamma fuctio by substitutig y m ξ Ω ad dξ Ω mξ dy i eq. (6.6). This results i, F X () Γ(m) mξ Ω m ξ e m ξ Ω dξ Γ(m) m Ω mξ Ω Γ(m) y m e y y dy m Ω m Ω Γ(m) y m e y dy m e m ξ Ω dy ( mξ Ω) Γ m Ω (m) Γ(m) (6.7) Sice Γ m Ω (m) Γ(m) as we have F X ( ).