# statistics

The Gamma distribution Gamma(πΌ,π½) with paramters πΌ>0 , and π½>0 is defined by the density

ππΌ,π½(π₯)=π½πΌΞ(πΌ)π₯πΌβ1πβπ½π₯,for allπ₯β₯0.
The Ξ function is defined by

Ξ(π )=β«β0π₯π β1πβπ₯ππ₯.
As usual, the constant π½πΌΞ(πΌ) is a normalization constant that gives β«β0ππΌ,π½(π₯)ππ₯=1.

In this problem, let π1,β¦,ππ be i.i.d. Gamma variables with

π½=1πΌfor some πΌ>0.
That is, π1,β¦,ππβΌGamma(πΌ,1πΌ) random variables for some πΌ>0 . The pdf for ππ is therefore

ππΌ(π₯)=1Ξ(πΌ)πΌπΌπ₯πΌβ1πβπ₯/πΌ,for all π₯β₯0.

What is the limit, in probability, of the sample average πβ―β―β―β―β―π of the sample in terms of πΌ ?

πβ―β―β―β―β―πββββπββπ = ?

----------------------------------------------------------------------------------------------------

Use the result from the previous problem to give a consistent estimator πΌΜ of πΌ in terms of πβ―β―β―β―β―π .

(Enter barX_n for πβ―β―β―β―β―π )

πΌΜ =?

----------------------------------------------------------------------------------------------------
For the Delta method to apply, at what value of π₯ does π need to be continuously differentiable? (Your answer should be in terms of πΌ .)

π₯=?
----------------------------------------------------------------------------------------------------

What is its asymptotic variance of πΌΜ ?

π(πΌΜ )=

1. 👍 4
2. 👎 0
3. 👁 529
1. Using the previous part, find confidence intervals for πΌ with asymptotic level 90% using both the βsolving" and the βplug-in" methods. Use π=25 , and πβ―β―β―β―β―π=4.5 .
(Enter your answers accurate to 2 decimal places. Use the Gaussian estimate π0.05β1.6448 for best results.)

Isolve=[ , ]
Iplug-in=[ , ]

1. 👍 3
2. 👎 0
2. (a) alpha^2
(b) barX_n^(1/2)
(c1) alpha^2
(c2) Normal Distribution
(c3) ?
(d)?

1. 👍 6
2. 👎 0
3. c1 answer above is wrong

1. 👍 0
2. 👎 4
4. I got green tick with alpha^2 for c1

1. 👍 3
2. 👎 0
5. c3 and d?

1. 👍 1
2. 👎 0

## Similar Questions

1. ### Maths: Partial Differentiation

1) If x = rcos theta and y = r sin theta, show that partial r / partial x = cos theta and find partial theta / partial x. 2) If z = sin theta.sin phi.sin gamma, and z is calculated for the values theta = 30degrees, phi = 45

2. ### Science

A scientist studying the effects of gamma rays on marigolds gathered the data shown below. Hypothesis: Marigolds exposed to higher levels of radiation will not grow as well as marigolds that are exposed to less radiation Plant A:

3. ### Science

A scientist studying the effects of gamma rays on marigolds gathered the data shown below. Hypothesis: Marigolds exposed to higher levels of radiation will not grow as well as marigolds that are exposed to less radiation Plant A:

4. ### science

Find the frequency and wavelength of a 50-MeV gamma-ray photon.

1. ### Bio

of the following, the best conclusion concerning the difference between the S phases for beta and gamma is that: a. gamma contains more DNA than beta b. beta and gamma contains the same amout of DNA c. beta contains more RNA than

2. ### Chemistry/Physics

"Colbalt-60 is a radioactive isotope used to treat cancers of the brain and other tissues. A gamma ray emitted by an atom of this isotope has an energy of 1.33 MeV (million electron volts; 1 eV = 1.602*10^-19 J). What is the

3. ### Math

Alpha writes the infinite arithmetic sequence 10, 8, 6, 4, 2, 0... Beta writes the infinite geometric sequence 9, 6, 4, 8/3, 16/9,... Gamma makes a sequence whose n^th term is the product of the n^th term of Alpha's sequence and

1. A Geiger counter measures radiation indirectly by measuring flashes of light electrons released by ionization speaker static curies answer: electrons released by ionization 2. Reactor control rods are made of a substance that

1. ### Chemistry

3. The three major types of radioactive decay of an unstable nucleus are alpha particles, beta particles, and gamma rays. Compare and contrast alpha particles, beta particles, and gamma rays. Explain how alpha decay works and how