13. The number of times an individual contracts a cold in a given year is a Pois

Question

13. The number of times an individual contracts a cold in a given year is a Poisson random variable with parameter -2. Suppose that a new wonder drug has just been mar- keted that reduces the Poisson parameter to -1 for 70% of the population. For the other 30% of the population the drug has no appreciable effect on colds. Given that an individual tries the drug for a year and has 0 colds in that time, how likely is it that the drug is beneficial for him/her (that is, the drug was effective with a Poisson rate = 1)? (Define all notation, including all events and any and all random variables.)

Explanation / Answer

Ans:

Let's call B the event that the drug is beneficial to someone (nonB the event that it is not), and C the number of colds contracted over one year.
Poisson distribution: P(X=k) = k*(e-/k! )

you know that P(C=0|B) = 10*e^(-1)/0!= e-1= 0.3679

P(C=0|nonB)= 20*e^(-2)/0!= e-2 = 0.1353

You must calculate P(B|C=0)

You know (Bayes' theorem) that:
P(B|C=0) = P(C=0|B)P(B)/P(C=0)

P(B) = 0.70(the drug is beneficial for 70% of those who try it)
P(C=0) = P(C=0|B)P(B) + P(C=0|nonB)P(nonB) (law of total probability)

P(C=0) = 0.3679*0.70 + 0.1353*0.30 = 0.2981

finally, P(B|C=0) = 0.3679*0.70/0.2981 = 0.8639

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.