Question

Let the mean success rate of a Poisson process be 12 successes
per hour.

**a.** Find the expected number of successes in a 19
minutes period. **(Round your answer to 4 decimal
places.)**

**b.** Find the probability of at least 2 successes in
a given 19 minutes period. **(Round your answer to 4 decimal
places.)**

**c.** Find the expected number of successes in a two
hours 30 minutes period. **(Round your answer to 2 decimal
places.)**

**d.** Find the probability of 23 successes in a given
two hours 30 minutes period. **(Do not round intermediate
calculations. Round your final answer to 4 decimal
places.)**

Answer #1

X ~ Poisson ()

Where = 12 successes per hour.

Poisson probability distribution is

P(X) = e^{-}^{X} /
X!

a)

E(X) = 12* 19 / 60 = **3.8**

b)

P(X >= 2) = 1 - P(X <= 1)

= 1 - [ P(X = 0) + P(X = 1) ]

= 1 - [e^{-3.8} + e^{-3.8} * 3.8 ]

= **0.8926**

c)

In 2 hour 30 minutes, there are 2 * 60 + 30 = 150 minutes.

E(X) = 12 * 150 / 60 = **30**

d)

P(X = 23) = e^{-30} * 30^{23} / 23!

= **0.0341**

Assume a Poisson random variable has a mean of 4 successes over
a 128-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 60 minutes. (Round intermediate
calculations to at least 4 decimal places and final answer to 4
decimal places.)

Assume a Poisson random variable has a mean of 8 successes over
a 128-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 55 minutes. (Round intermediate calculations to at least 4
decimal places and final answer to 4 decimal places.)

Telephone calls arrive at a 911 call center at the rate of 12
calls per hour.
What is the mean number of calls the center expects to receive
in 30 minutes?
What is the standard deviation of the number of calls the center
expects to receive in 30 minutes? (Round your answer to two decimal
places.)
Find the probability that no calls come in for 30 minutes.
(Round your answer to five decimal places.)

Suppose small aircraft arrive at a certain airport according to
a Poisson process with rate α = 8 per hour, so that the number of
arrivals during a time period of t hours is a Poisson rv with
parameter μ = 8t. (Round your answers to three decimal places.)
(a) What is the probability that exactly 5 small aircraft arrive
during a 1-hour period?
What is the probability that at least 5 small aircraft arrive
during a 1-hour period?
What...

2. The arrival of insurance claims follows a Poisson
distribution with a rate of 7.6 claims per hour.
a) Find the probability that there are no claims during 10
minutes.
b) Find the probability that there are at least two claims
during 30 minutes.
c) Find the probability that there are no more than one claim
during 15 minutes.
d) Find the expected number of claims during a period of 2
hours. (5)

Assume a Poisson random variable has a mean of 10 successes over
a 120-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?
c. Find the probability that the time to
success will be more than 54 minutes

2. The arrival of insurance claims follows a Poisson
distribution with a rate of 7.6 claims per hour.
a) Find the probability that there are no claims during 10
minutes. (10)
b) Find the probability that there are at least two claims
during 30 minutes. (10)
c) Find the probability that there are no more than one claim
during 15 minutes. (10)
d) Find the expected number of claims during a period of 2
hours. (5)

The exponential distribution is frequently applied
to the waiting times between successes in a Poisson
process. If the number of calls received per hour
by a telephone answering service is a Poisson random
variable with parameter λ = 6, we know that the time,
in hours, between successive calls has an exponential
distribution with parameter β =1/6. What is the probability
of waiting more than 15 minutes between any
two successive calls?

In the binomial probability distribution, let the number of
trials be n = 3, and let the probability of success be p = 0.3742.
Use a calculator to compute the following.
(a) The probability of two successes. (Round your answer to
three decimal places.)
(b) The probability of three successes. (Round your answer to
three decimal places.)
(c) The probability of two or three successes. (Round your
answer to three decimal places.)

Let the probability of success on a Bernoulli trial be 0.21. a.
In seven Bernoulli trials, what is the probability that there will
be 6 failures? (Do not round intermediate calculations. Round your
final answer to 4 decimal places.) b. In seven Bernoulli trials,
what is the probability that there will be more than the expected
number of failures? (Do not round intermediate calculations. Round
your final answer to 4 decimal places.)

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