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 X_{1} denotes the number of individuals who developed mild side effects and X_{2} denotes the number of individuals who developed severe side effects, then the coefficient of variation of X_{1}+ X_{2} 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

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