A test for the presence of a certain disease has probability 0.4 of giv- ing a f

Question

A test for the presence of a certain disease has probability 0.4 of giv- ing a false positive reading and has probability 0.15 of giving a false negative reading. Suppose 8 individuals are tested, four of whom have the disease and four of whom do not. Let Y = the number of positive readings in the 8 people. (a) Does Y have a binomial distribution? Explain your reasoning. Also consider the random variables Xi = the number of positive readings in the 4 people with the disease and X2 - the number of positive readings in the 4 people without the disease. Do Xi and (b) Calculate P (Y = 2). (Hint: it may be helpful to write Y in terms of Xi and X2, and consider all pairs (xi, T2) which result in the desired outcome.)

Explanation / Answer

a) as probability of having positive reading is differnet for people having disease or not,

there probability is not independent of trail and is not fixed therefore it is not binomial distribution

.X1 and X2 seperately have binomial distribution as nuumber of trails in both group of people is fixed and with in each group  probability of having positive  reading is independent.

b)here for X1 ; p1 =0.85 and n1 =4 and for X2 ; p2 =0.4 and n2 =4

P(Y=2) =P(X1=0,X2=2)+P(X1=1; X2=1)+P(X1=2,X2=0)

=(4C2)*(0.85)2(0.15)2*(4C0)*(0.4)0(0.6)4 +  (4C1)*(0.85)1(0.15)3*(4C1)*(0.4)1(0.6)3 +(4C0)*(0.85)0(0.15)4*(4C2)*(0.4)2(0.6)2 = 0.0168

lease revert for any calrifcation required

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