Question
Asked Aug 8, 2019
27 views

Suppose that Xi ∼ Gamma(αi , β) independently for i = 1, . . . , N. The mgf(moment generating function) of Xiis MXi(t) = (1 − (t/β) )−αi . (a)Use the mgf of Xi to derive the mgf of ∑i=1 Xi .

Use the mgf of Xi to derive the mgf of ∑i=1 Xi

check_circle

Expert Answer

Step 1

Introduction:

It is given that Xi’s (i = 1, 2, …, N) are independent Gamma (αi, β) random variables with mgf as follows:

help_outline

Image Transcriptionclose

M0- 1 B

fullscreen
Step 2

Derivation:

The mgf of the sum of all Xi’s, that is, of Σ Xi is calculated as follows:

help_outline

Image Transcriptionclose

M()M+Xx ,(e) =Mx (t) My (t).. .M(t) X, are independent] B -(a+a2+..ay = mgf of Gamma (>a.B)

fullscreen
Step 3

Conclusion:

The above derivation shows that Σ Xi has the Gamma (Σ α...

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Statistics

Other

Related Statistics Q&A

Find answers to questions asked by student like you
Show more Q&A
add
question_answer

Q: Assume that we have two normal distributions both with means 0 and standard deviations σ1 and σ2. If...

A: We are given two normal distributions with mean = 0 and standard deviations σ1 and σ2.

question_answer

Q: In a recent survey it was stated that Americans watch television on average four hours per day. Assu...

A: Hey, since there are multiple subparts posted, we will answer first three subparts. If you want any ...

question_answer

Q: Fuming because you are stuck in traffic? Roadway congestion is a costly item, both in time wasted an...

A: Hey, since there are multiple subpart questions posted, we will answer first three subparts. If you ...

question_answer

Q: A. A population has a mean of 6.9 and standard deviation of 2.6. Find the value that has 80% of the ...

A: Given Population mean = 6.9Standard deviation = 2.6The Z score with the area left to it close to 0.8...

question_answer

Q: Compute the probability of an event E if the odds in favor of E are 2/5.

A: Odds of an event:The odds an event is given by the ratio of the probability of the event to its prob...

question_answer

Q: A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters...

A: (a)Poisson distribution is most suitable, when the events are “rare” and occur independently of each...

question_answer

Q: I am working on constructing a histogram for some data for the 39 US national parks under 900,000 ac...

A: According to the given data41 ,66 ,233 ,775 ,169, 36 ,338 ,233 ,236, 64,183 ,61 ,13 ,308 ,77,520, 77...

question_answer

Q: In a normal distribution with mean 3 and variance 49, what are the upper and lower limit scores for ...

A: The given information is that the scores distributed normally with mean 3 and variance is 49. The st...

question_answer

Q: A personnel director in a particular state claims that the mean annual income is the same in one of ...

A: According to the given question, in order to test that the annual income is the same in one of the​ ...