Suppose that the autopilot program in a plane makes an average of one error ever

Question

Suppose that the autopilot program in a plane makes an average of one error every
4 hours, and the number of errors made in a given time period follows a Poisson
distribution. Note that if the number of errors made in a one hour period is modeled
by a Poisson() distribution, then the number of errors made in a k hour period is
modeled by a Poisson(k ) distribution.
(a) What is the probability that the autopilot program makes no errors during a 10
hour
ight? State the fully specied distribution you are using along with the
probability.
(b) What is probability that the autopilot program makes any number of errors
(greater than 0) during a 10 hour
ight?
(c) Suppose we are about to take a four hour
ight. State the fully specied distri-
bution for the number of errors made during the four hour
ight along with the
expected number of errors and the variance for the number of errors.
(d) Suppose two
ights are leaving an airport at the same time (assume they are
independent). One will be making an 8 hour
ight, the second will be making a
4 hour
ight. What is the probability that together, the two autopilot programs
make less than 2 errors? State the fully specied distribution you are using as
well as the probability.

Explanation / Answer

Given X~Poisson(=1 per 4 hours)

X~Poisson(=2.5 per 10 hours)

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(a) What is the probability that the autopilot program makes no errors during a 10
hour ight?

P(X=0)=exp(-2.5) = 0.082

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(b) What is probability that the autopilot program makes any number of errors
(greater than 0) during a 10 hour ight?

P(X>0)=1-P(X=0)=1-0.082 = 0.918

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