At Oxford University, testing for covid-19 has been carried out on campus on a large scale this spring. For each test taken, it is recorded whether it is positive or negative. The probability that a random student who tests himself on campus tests positive is p. Let X be the number of tests performed even if one has found a positive test for the first time.

a) Explain why X is given a point probability

P(X=k) = (1p)^k-1p, k= 1,2, ...

b) The moment generating function of X is given by

M_X(t) =e^tp/1(1p)e^t

Derive expressions for E (X) and V (X) and find the expected number of tests to and with first positive test if p = 0.001.

Now let X₁ be the number of tests up to and including the first positive, X₂ the number of tests from the first positive up to and including the second positive, and further X_r the number of tests from (r 1) positive up to and including r positive. Then Y = ^r_i=1 Xi number of tests up to and including r positive test.

c) Find E (Y) expectation and V (Y) variance and calculate the expected number of tests performed up to and including the 10th positive, if p is still 0.001.

d) Find expression for P(Y=k), k = r, r + 1, ....

e) The test is not accurate. For the rapid test used on campus, studies show that the probability of testing positive, given that you are infected, is only 0.75. At the same time, we know that the test can give false positives. The probability that the test is positive, even if you are not infected, is 0.0001. We still assume that the probability is p = 0.001 that a student tests positive. Show that the probability that a random student testing on campus is actually infected is 0.0012.

f) Find the probability that a random student who tests himself on campus is infected, given that the test is positive.