The number of times that a person contracts a cold in a given year is a Poisson
ID: 3203631 • Letter: T
Question
The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter lambda = 5. Suppose that a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to A = 3 for 75 % of the population. For the other 25% of the population, the drug has no appreciable effects on colds. If an individual tries the drug for a year and has 3 colds in that time, how likely is it that the drug is NOT beneficial for her/him? (Let X be the number of colds in that year, B = {the drug is beneficial}. Compute P(B^c|X = 3)).Explanation / Answer
Lets say 100 people were picked
The wonder drug has an effect on 75 people
It doesn't work for 25 people
When the drug has effect lambda=3
When it has no effect lambda=5
X=Having 3 colds per year
P(B)=Probability of drug being effective = 0.75
P(Bc)=Probability of the drug not being effective = 0.25
Poisson Probability= P(X|m)= (e-m)*(mX)/ X!
Where
P(X|B)= P(3 colds) for 75 people= (e-3)*(33)/3! = 0.224
P(X|Bc)= P(3 colds) for 25 people=(e-5)*(53)/3! =0.1403
According to Baye's theorm, Probability of drug being not benificial is
P(Bc|X=3) = { P(X|Bc)P(Bc) } / { P(X|Bc)P(Bc)+P(X|B)P(B)
= 0.0351/(0.0351+0.168)
=0.1728
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