Solved: The Gamma distribution is a distribution that arises naturally in proc

Home
questions and answers
the gamma distribution is a distribution that arises naturally in proc

Question

Statistics For Business And Economics

The Gamma distribution is a distribution that arises naturally in processes for which the waiting times between events are relevant. It can be thought of as a waiting time between Poisson distributed events, and is used, for example, to model the amount of rainfall accumulated in a reservoir, or the size of loan defaults or aggregate insurance claims. The gamma distribution is a two-parameter exponential family with two natural parameters (scale and shape). The density of a random variable Y with Gamma(a, v) distribution,a > 0, v, y < ∞, can be written as

f_{Y}(y)=\frac{y^{\nu-1} \alpha^{\nu} e^{-y \alpha}}{\Gamma(\nu)}

where I'(v) is a value of a known function that ensures that fy integrates to 1.We hold fixed v, the shape parameter.

(a) Show that Gamma distribution with fixed v is a member of the one-parameter exponential family of distributions. [5 marks]

(b) What are the canonical and dispersion parameters for the gamma distribution? Determine the variance function. [5 marks]

Answer

Verified

Question 45696

Statistics For Business And Economics

(1 point) A biotechnology company produced 182 doses of somatropin, including 12 which were defective. Quality control test 14 samples at random, and rejects the batch if any of the random samples are found defective. What is the probability that the batch gets rejected?

Statistics For Business And Economics

(1 point) Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is 2.2. What is the probability that there will be
(a) at most 5 such accidents in the next month?
(b) less than 4 such accidents in the next 3 months?
(c) exactly 8 such accidents in the next 5 months?

Statistics For Business And Economics

(1 point) A certain typing agency employs two typists. The average number of errors per article is 1 when typed by the first typist and 3.2 when typed by the second. If your article is equally likely to be typed by either typist, find the probability that it will have no errors.

Statistics For Business And Economics

(1 point) The mean number of patients admitted per day to the emergency room of a small hospital is 2. If, on any given day, there are only 7 beds available for new patients,what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients?

Statistics For Business And Economics

nt) Given that x is a random variable having a Poisson distribution, compute the following:
\text { (a) } P(x=7) \text { when } \mu=0.5
P(x \leq 6) \text { when } \mu=4.5
\text { c) } P(x>3) \text { when } \mu=5.5
P(x<9) \text { when } \mu=4.5

Statistics For Business And Economics

B. Compute the probability that in the next 8 hours, more than 24 cars will arrive.
A. Determine the probability that over the next hour, only one car will arrive.
Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 9 per hour.

Statistics For Business And Economics

Complaints about an Internet brokerage firm occur at a rate of 5 per day. The number of complaints appears to be Poisson distributed.
A. Find the probability that the firm receives 4 or more complaints in a day.
Probability =%3D
B. Find the probability that the firm receives 22 or more complaints in a 5-day period.
Probability =

Statistics For Business And Economics

Flaws in a carpet tend to occur randomly and independently at a rate of one every 150 square feet. What is the probability that a carpet that is 9 feet by 13 feet contains noflaws?
Probability:

Statistics For Business And Economics

The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.4 per week. Find the probability of the following events.
.4 or more accidents occur in a week.
A. No accidents occur in one week.
One accident occurs today.

Statistics For Business And Economics

(1 point) An airline company is considering a new policy of booking as many as 271 persons on an airplane that can seat only 260. (Past studies have revealed that only 89%of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 271 persons. not enough seats will be available.