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Assume the number of episodes per year of a disease follow Poisson distribution with parameter $u=1.6$ per year.

1) What is the probability that two siblings will both have three or more episodes of disease in the first two years of life?

ans: $u=1.6^2\text{ (times 2 because 2 year)}=3.2$, so $P(X\ge3) \cdot P(X\ge3) = [1-P(X=0)-P(X=1)-P(X=2)]^2$ by applying $u=3.2$

2) what is the probability that exactly one siblings will have three or more episodes of disease in the first two years of life?

ans: $u=1.6^2\text{ (times 2 because 2 year)}=3.2$, so $P(X\ge3)= 1-P(X=0)-P(X=1)-P(X=2)$ by applying $u=3.2$

3) what is the expected number of siblings, in a 2-sibling family, who will have three or more episodes in the first two years of life?

May anyone help me solve the question 3? help me check whether question 1 and 2 are correct?


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noob: Changed $P$ to $X$ thrice in your post. Tell me if this is OK. – Did Sep 27 '11 at 16:02

Your first answer looks alright, except that you've written $1.6^2$ where I'm guessing you meant $1.6\cdot2$. You can reduce the whole thing to an actual number by realizing that $P(X=0) = e^{-3.2}$, $P(X=1)=3.2e^{-3.2}$, and $P(X=2)=3.2^2 e^{-3.2}/2$.

Your second answer is wrong. What you need is the probability that the first sibling has at least three such episodes and the second does not or the second one does and the first does not.

For the third question, you need the answers to the first two questions and also the probability that neither sibling has three or more such episodes. Then the expected number of siblings having three or more such episodes is $$ \begin{align} & \phantom{.}\qquad 0\cdot P(\text{neither sibling has three or more episodes}) \\ & {}\ {} + 1\cdot P(\text{exactly one sibling has three or more episodes}) \\ & {}\ {}+ 2\cdot P(\text{both siblings have three or more episodes}). \end{align} $$

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I wonder why the word "sibling" was used in the first question. If a person (child?) has illnesses that have a Poisson distribution, then the assumption that a sibling's illnesses have the same distribution and are independent of the person's illnesses is open to question; there may be a high degree of correlation in the illnesses of siblings, especially if infectious diseases are involved. – Dilip Sarwate Sep 27 '11 at 18:38
@Dilip: If you pick a pair of siblings randomly out of a population, then both members of one pair of siblings may have similar degrees of susceptibility far more often than would non-siblings so chosen. So they'd be positively correlated. BUT they might yet be CONDITIONALLY independent GIVEN the value of $u$ in the particular case. The value of $u$ may differ from one pair of siblings to the next, and that's where the correlation would come from. On the other hand, if the disease is communicable, they might be correlated even given the particular value of $u$. – Michael Hardy Sep 28 '11 at 2:40

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