Question

In: Statistics and Probability

Assume a Poisson random variable has a mean of 9 successes over a 126-minute period. a....

Assume a Poisson random variable has a mean of 9 successes over a 126-minute period.

a. Find the mean of the random variable, defined by the time between successes.

b. What is the rate parameter of the appropriate exponential distribution? (Round your answer to 2 decimal places.)

c. Find the probability that the time to success will be more than 59 minutes. (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.)

Expert Solution

Answer:

Given that:

Assume a Poisson random variable has a mean of 9 successes over a 126-minute period.

a) Find the mean of the random variable, defined by the time between successes.

It is given that, a Poisson random variable has a mean of 9 successes over a 126 minute period.

That is

Since the successes follow Poisson distribution, the time between successes has an exponential distribution.

The mean of the random variable defined by the time between successes is

Thus,the mean of the random variable, defined by the time between successes is 14 minutes

b) What is the rate parameter of the appropriate exponential distribution?

In exponential distribution, is the rate parameter which is the inverse of the mean

The required rate parameter is

Thus, the rate parameter of the appropriate exponential distribution is 0.07

c) Find the probability that the time to success will be more than 59 minutes

The cumulative distribution function of Exponential distribution is

The probability that the time to success wil be more than 59 minutes is

Thus,the probability that the time to success will be more than 59 minutes is 0.0161

orchestra answered 1 year ago

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