## JAM 2022 [ 51-60 ]

A vaccine, when it is administered to an individual, produces no side effects with probability 4/5 , mild side effects with probability 2/15 and severe side effects with probability 1/15. Assume that the development of side effects is independent across individuals. The vaccine was administered to 1000 randomly selected individuals. If X1 denotes the number of individuals who developed mild side effects and X2 denotes the number of individuals who developed severe side effects, then the coefficient of variation of X1+ X2 equals _________________. (round off to 2 decimal places)

Clearly,

(X_1,X_2) \sim Multinomial \left(n=1000,p_1= \frac{2}{15},p_2=\frac{1}{15} \right)

And so,

E(X_1) = np_1=800  , V(X_1) = np_1(1-p_1) \\
E(X_1)=np_2, V(X_2)=np_2(1-p_2) \\
cov(X_1,X_2) =-np_1p_2
E(X_1+X_2) = n(p_1+p_2) = \frac{3000}{15}=200 \\
V(X_1+X_2) = np_1(1-p_1)+np_2(1-p_2) -2 np_1p_2 \\
= n \left[ (p_1+p_2) - (p_1+p_2)^2  \right] \\
= 1000 \left[ \frac{1}{5} + \frac{1}{25}  \right] \\
= 1000 \left[ \frac{6}{25}  \right] \\
=240

CV =  \frac{\sqrt{V(X_1+X_2)}}{E(X_1+X_2)} = \sqrt{240}/200 = 0.0774