We administer a vaccine against a virus with the following rules. a-) If the first dose provides a high concentration of antibodies, we stop at it. This >happens with probability $\frac{30}{100}$. In this case the person has protection for 2 >months. b-) If the first dose does not provide a high concentration of antibodies, we administer >a second dose. The second dose can provide a high antibody concentration with >probability $\frac{60}{100}$ which means a person has protection for 4 months or low >antibody concentration which means a person has protection for 3 months. Calculate the expectation and variance of the number of months.
1st case
The first dose works 0.3 times and guarantees 2 months of protection.
2nd case
The first dose fails 0.7 of the time and we get great concentration in the second in 0.6, that is, in 0.42 of the times and guarantees 4 months of protection.
3rd case
The first dose fails 0.7 of the time and we get low concentration in the second by 0.4, that is, 0.28 of the time and guarantees 3 months of protection.
$E[X] = 0.3*2 + 0.42*4 + 0.28*3 = 3.12$ months
$E[X^2] = 0.3*2^2 + 0.42*4^2 + 0.28*3^2 = 10.44$
$Var = E[X^2] – (E[x])^2 = 10.44 – 3.12^2 = 10.44 – 9.73 = 0.71$
Is this correct?
Thaks for any help.
Yes, it is.
Those are the formula to use, and the numbers to use in them.
The calculations check out too (when rounded to the second decimal place).