Patients given drug therapy either improve remain the same o

Patients given drug therapy either improve, remain the same, or degrade with probabilities 0.5, 0.4,
0.1, respectively. Suppose that 20 patients (assumed to be independent) are given the therapy. Let X1, X2
and X3 denote the number of patients who improved, stayed the same, or became degraded. Determine the
following:

(a) Are X1, X2, X3 independent?
(b) P(X1 = 10)
(c) P(X1 = 10, X2 = 8, X3 = 2)
(d) P(X1 = 5|X2 = 12)
(e) E(X1)

Solution

n =20

x1,x2 and x3 are independent.

this is binominal disteibution where P(x=r) = nCr * p^r * q^(n-r)

P(x1=10)

p = 0.5 and q = 0.5

P(x1=10) = 20C10 * 0.5^10 * 0.5^10 = 0.1762

2.

P(x1=10,x2=8,x3=2) = P(x1=10) * P(x2=8) * P(x3=2)

P(x1=10,x2=8,x3=2) = 20C10 * 0.5^10 * 0.5^10 * 20C8 * 0.4^8 * 0.6^12 20C2 * 0.1^2 * 0.9^18

P(x1=10,x2=8,x3=2) = 0.00903

3.

P(x1=5|x2=12) = P(x1=5)/P(x2=12)

P(x1=5|x2=12) = ( 20C5 * 0.5^5 * 0.5^15) / (20C12 * 0.4^12 * 0.6^8)

P(x1=5|x2=12) = 0.4165

4.

E(x1) = n*p = 20 * 0.5 = 10