Question

3. Let the random variable X be the number of days that a patient needs to be in hospital. Suppose X has p.f. f(x) =k(6-0) x = 1,2,3,4,5. (a) Find the value of the multiplicative constant k, so that f(x) is a probability function. (b) What is the expected length of stay? If you cannot answer (a), write this in terms of k. (c) Suppose the patient has medical insurance, and they are to receive $200 for each of the first two days in hospital, and $100 for each day after the first two days, what is the expected payout for the hospitalisation? If you cannot answer (a), write this in terms of k.

Answer 1

Suppose. x has pmif 7 (2) = K (6-x) x=1,2,3,4,5 Then E f(x) = 1 => [(6-1)+(6-2) + (6-3) + (6-4) + (6-5)] = 1 - k [5+4+3+3+1] = 1 = = 3/45. is The p. i f. x is f(x) = 2 (6-x). 6) Expected length of stay = E(x) = {x} (x) = Z 18 x (02). T5 [1.5+2.4+3.3+4.2 +5.1] = 30 15 = 7/. c) Expected Payout for the hospitalisation = $200 F(x) + $200 / (22) +$100 [](23) + f (x2)+7 (25) 8$ 168[ 200(6-1) + 200.66-2)4 100.6-3+100-(6-304160(65) - $ 5 (1000+800 +.300 + 200 +100) = $ 15 x 2400 $160.