Question

A bag contains 4 red and 5 green balls. Let X denotes number of red and Y denotes number of green balls in 2 drawn from the bag by random. Find joint probability distribution, compute E(x), E(Y) and correlation coefficient. 2. A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities: (a) That the test result will be positive; (b) that, given a positive result, the person is a sufferer; (c) that, given a negative result, the person is a non-sufferer; (d) That the person will be misclassified. Mth251 Pag

Question 1. (20 pts.) 1. A bag contains 4 red and 5 green balls. Let X denotes number of red and Y denotes number of green balls in 2 drawn from the bag by random. Find joint probability distribution, compute E(X), E(Y) and correlation coefficient. 2. A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities: (a) That the test result will be positive; (b) that, given a positive result, the person is a sufferer; (c) that, given a negative result, the person is a non-sufferer; (d) That the person will be misclassified.

Answer 1

draw a balls we rand only drow only x+y=2 will give probatality nonzero case-t when Green balls is drawn Y 1 2 B4) Case (2) when red ball is doaulos ulos shoo 19 х 0 I 5118 2. P(x) 519 116 Joint probability x 14 6. 1 2. O 0 0 5118 1 O واک o z a 116 comelatin coffcrento Now vi Elx) ons .819 ECU? EX) sos + 1x 3 + 2x / &lg 61181 8 + 1*9 / +421 / oks 18 =1l/g Ey = Ox} +245 +2x = 119 Gy? = 121 121

E(42) = Oxft last t 4 X 5 = iB w.ro ERIV) - 519 As we know that Cow (x,y) = E(X,Y) – Ex) EU) 819x 111g 519 = -U3 8) On i E(X2) - E(X) 2 11/9 -64181 = 35 81 og EM2) - E(4)? 121 81 - 14181 Correlation (officient = Cor(x,y) (Ency _U3181 = 38xlu 81 81 Carrelection coff = -7.133 • 2. propability of pusitue result = 0.95 when applied to suffering person when applied to non sufferer I probability of false result 0.10 0.51 population suffers,

(a) Probability of test being positive Pas = test postive when pressen suffers + 'false result obred person is non sufenses probability of the presson being sufferes 0.57 = 0005 WN PiA = 0.005*0.95 + 0,995 x 0.10 6 10.10425 given a positue result, the presson is suffere. postue result when pesson euffers positive result when person sufferst false report PCB) = 0.005*0.95 0.005x0.95 +0.995X0-10 =10.04556 ☺ given negative result when pessons Is non suffer Perregal he result when person is nen suffer) PC) = negative result) probabilty 0.995 X 6.90 0.99972 0.995X0.90 +0.00561-0.95) probability of being the classified PCsuffere and negathereault) + Plnom suffere + postueretung 0-005*(1-0.959 +0.995*0.10 - - (d) Z z 0.09975