A person with a cough is a persona non grata on airplanes, elevators, or at the

Question

A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters especially, the irritation level rises with each muffled explosion. According to Dr. Brian Carlin, a Pittsburgh pulmonologist, in any large audience you'll hear about 12 coughs per minute.

(a) Let r = number of coughs in a given time interval. Explain why the Poisson distribution would be a good choice for the probability distribution of r.

Coughs are a common occurrence. It is reasonable to assume the events are independent.Coughs are a rare occurrence. It is reasonable to assume the events are independent.     Coughs are a rare occurrence. It is reasonable to assume the events are dependent.Coughs are a common occurrence. It is reasonable to assume the events are dependent.

(b) Find the probability of six or fewer coughs (in a large auditorium) in a 1-minute period. (Use 4 decimal places.)


(c) Find the probability of at least six coughs (in a large auditorium) in a 26-second period. (Use 4 decimal places.)

Explanation / Answer

a.
Let r = number of coughs in a given time interval
the Poisson distribution would be a good choice for the probability distribution of r
because Coughs are a common occurrence. It is reasonable to assume the events are independent
thats why good choice is Poisson distribution.
b.
POSSION DISTRIBUTION
pmf of P.D is = f ( k ) = e- x / x!
where
= parameter of the distribution.   
x = is the number of independent trials
I.
mean =
= 12

the probability of six or fewer coughs (in a large auditorium) in a 1-minute period
P( X < = 6) = P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-12 * 12 ^ 6 / 6! + e^-12 * 3 ^ 5 / 5! + e^-12 * ^ 4 / 4! + e^-12 * ^ 3 / 3! + e^-12 * ^ 2 / 2! + e^-12 * ^ 1 / 1! + e^-12 * ^ 0 / 0!
= 0.04582 =0.0458
c.
the probability of at least six coughs (in a large auditorium) in a 26-second period
mean = = 12 per minute
here 26 seconds so that = 12*(26/60) =5.2 per 26 seconds
P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-5.2 * 3 ^ 5 / 5! + e^-5.2 * ^ 4 / 4! + e^-5.2 * ^ 3 / 3! + e^-5.2 * ^ 2 / 2! + e^-5.2 * ^ 1 / 1! + e^-5.2 * ^ 0 / 0!   
= 0.58091,
P( X > = 6 ) = 1 - P (X < 6) = 0.41909 =0.4191

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

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