Math 309

Gamma Distribution

In a Poisson Process, the length of time before the event occurs t times has a Gamma distribution with parameters ( t, where  is the mean of the Poisson distribution.

Example: Suppose that, on average, the number of-particles emitted from a radioactive substance is four every second. What is the probability that it takes at least two seconds before the next three -particles are emitted?

Let N(t*) denote the number of -particles emitted from a radioactive substance in the time interval [0, t*], t*>0. It is reasonable that N(t*) is a Poisson process with when t* = 1. X, the time between now and the third -particle is emitted, has a gamma distribution. We want .

Exercises.

1. Show that (t) = (t-1)(t-1).

  1. The response times on an online computer terminal have approximately a gamma distribution with mean 4 seconds and variance 8 seconds squared. Write the probability density function for the response time. What is the probability that the response time would exceed 5 seconds?

3. Customers arrive at a restaurant at a Poisson rate of 12 per hour.

a) What is the probability that at least two hours elapse before the 20th customer arrives?

b)If the restaurant makes a profit only after 30 customers have arrived, what is the expected length of time until the restaurant starts to make a profit?

4. What is the relationship between the Gamma distribution and the exponential distribution?

Beta Distribution

  1. The proportion of a brand of televisions sets requiring service during the first year of operation is a random variable having a beta distribution with a = 3 and b = 2.

a)What is the probability that at least 80% of the new models sold this year of this brand will require service during their first year of operation?

b)What is the expected proportion of television sets that will require service during the first year?

Weibull Distribution.

6. Resistors used in the construction of an aircraft guidance system have life lengths that follow a Weibull distribution with (with measurements in thousands of hours).

a) Find the probability that the life length of a randomly selected resistor of this type exceeds 5000 hours.

b) If three resistors of this type are operating independently, find the probability that exactly one of the three will burn out prior to 5000 hours use.

7. Is there a relationship between the Weibull distribution and the Exponential distribution?

Answers.

Example: 41e-8

  1. Use integration by parts.
  2. .1803b. 31/12 hrs. = 2 hrs. 35 minutes
  3. 3.5e-2.5
  4. Exponential is Gamma with = 1.
  5. (collected hw)
  6. (collected hw)
  7. Weibull with =1 is Exponential with 1/ = .