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.)

Solutions

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

Next > < Previous

Related Solutions

1. If the random variable x has a Poisson Distribution with mean μ = 53.4, find...

1. If the random variable x has a Poisson Distribution with mean μ = 53.4, find the maximum usual value for x. Round your answer to two decimal places. 2. In one town, the number of burglaries in a week has a Poisson distribution with mean μ = 7.2. Let variable x denote the number of burglaries in this town in a randomly selected month. Find the smallest usual value for x. Round your answer to three decimal places. (HINT:...

Let Y denote a random variable that has a Poisson distribution with mean λ = 6....

Let Y denote a random variable that has a Poisson distribution with mean λ = 6. (Round your answers to three decimal places.) (a) Find P(Y = 9). (b) Find P(Y ≥ 9). (c) Find P(Y < 9). (d) Find P(Y ≥ 9|Y ≥ 6).

(9) Assume on average 10 passengers arrive per minute. Assuming poisson arrivals and departures, estimate the...

(9) Assume on average 10 passengers arrive per minute. Assuming poisson arrivals and departures, estimate the gain (if any) in ‘average time spent in system per passenger’ if TSA decides to replace 4 type-A security scanners with 3 type-B security scanners. The service rate per scanner for type-A scanners is 3 passengers per minute and type-B scanners is 5 passengers per minute?

Assume X is a Poisson random variable with parameter lambda. What is the test statistic and...

Assume X is a Poisson random variable with parameter lambda. What is the test statistic and the best critical region for testing H_0:lambda = lambda_0 versus H_a:lambda = lambda_1 where lambda_0 < lambda_1 are constants?

X is a Gaussian random variable with variance 9. It is known that the mean of...

X is a Gaussian random variable with variance 9. It is known that the mean of X is positive. It is also known that the probability P[X^2 > a] (using the standard Q-function notation) is given by P[X^2 > a] = Q(5) + Q(3). (a) [13 pts] Find the values of a and the mean of X (b) [12 pts] Find the probability P[X^4 -6X^2 > 27]

The random variable X follows a Poisson process with the given mean. Assuming mu equals 6...

The random variable X follows a Poisson process with the given mean. Assuming mu equals 6 commaμ=6, compute the following. (a) P(66) (b) P(Xless than<66) (c) P(Xgreater than or equals≥66) (d) P(55less than or equals≤Xless than or equals≤77)

The number of customer arrivals at a bank's drive-up window in a 15-minute period is Poisson...

The number of customer arrivals at a bank's drive-up window in a 15-minute period is Poisson distributed with a mean of seven customer arrivals per 15-minute period. Define the random variable x to be the time (in minutes) between successive customer arrivals at the bank's drive-up window. (a) Write the formula for the exponential probability curve of x. (c) Find the probability that the time between arrivals is (Round your answers to 4 decimal places.) 1.Between one and two minutes....

Let p(y) denote the probability function associated with a Poisson random variable with mean λ. Show...

Let p(y) denote the probability function associated with a Poisson random variable with mean λ. Show that the ratio of successive probabilities satisfies (p(y)/p(y-1)) = ( λ /y), for y=1,2,...

The number of messages that arrive at a Web site is a Poisson random variable with...

The number of messages that arrive at a Web site is a Poisson random variable with a mean of five messages per hour. a. What is the probability that five messages are received in 1.0 hour? b. What is the probability that 10 messages are received in 1.5 hours? c. What is the probability that fewer than two messages are received in 0.5 hour? d. Determine the length of an interval of time such that the probability that no messages...

A binomial random variable with parameters n and p represents the number of successes in n...

A binomial random variable with parameters n and p represents the number of successes in n independent trials. We can obtain a binomial random variable by generating n uniform random numbers 1 2 n U ,U ,...,U and letting X be the number of i U that are less than or equal to p. (a) Write a MATLAB function to implement this algorithm and name the function gsbinrnd. You may use the for loop to generate random numbers. (b) Use...

Subjects