X₁, Instructions for Exercises 1 to 4: In each of these exercises, assume that the random variables Xn form a random sample of size n form the distribution specified in that exercise, and show that the statistic T specified in the exercise is a sufficient statistic for the parameter: Exercise 1: A normal distribution for which the mean μ is known and the variance ² is unknown; T = ₁₁ (X₁ -µ)². Exercise 2: A gamma distribution with parameters a and 3, where the value of 3 is known and the value of a is unknown (a > 0); T = I₁ X₁. Exercise 3: A uniform distribution on the interval [a, b], where the value of a is known and the value of b is unknown (b> a); T = max(X₁,..., Xn).
X₁, Instructions for Exercises 1 to 4: In each of these exercises, assume that the random variables Xn form a random sample of size n form the distribution specified in that exercise, and show that the statistic T specified in the exercise is a sufficient statistic for the parameter: Exercise 1: A normal distribution for which the mean μ is known and the variance ² is unknown; T = ₁₁ (X₁ -µ)². Exercise 2: A gamma distribution with parameters a and 3, where the value of 3 is known and the value of a is unknown (a > 0); T = I₁ X₁. Exercise 3: A uniform distribution on the interval [a, b], where the value of a is known and the value of b is unknown (b> a); T = max(X₁,..., Xn).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Solve Exercise 1,3,4,5
![Instructions for Exercises 1 to 4: In each of these exercises, assume that the random variables
X₁, Xn form a random sample of size n form the distribution specified in that exercise,
and show that the statistic T specified in the exercise is a sufficient statistic for the parameter:
Exercise 1: A normal distribution for which the mean is known and the variance o² is
unknown; T = ²₁ (X₁-µ)².
Exercise 2: A gamma distribution with parameters a and 3, where the value of 3 is known
and the value of a is unknown (a > 0); T = II/1 X₁.
Exercise 3: A uniform distribution on the interval [a, b], where the value of a is known and
the value of b is unknown (b> a); T = max(X₁, Xn).
..
7
Exercise 4: A uniform distribution on the interval [a, b], where the value of b is known and
the value of a is unknown (b> a); T = min(X₁,, Xn).
... "
Exercise 5: Suppose that X₁, Xn form a random sample from a gamma distribution
with parameters a > 0 and 3 > 0, and the value of 3 is known. Show that the statistic
T = log X, is a sufficient statistic for the parameter a.
Exercise 6: The Pareto distribution has density function:
f(x|xo,0) = 0xx-0-1, x ≥
Σ
0, 0>1
Assume that ro> 0 is given and that X₁, X2,, Xn is an i.i.d. sample. Find a sufficient
statistic for by (a) using factorization theorem, (b) using the property of exponential family.
Are they the same? If not, why are both of them sufficient?
Exercise 7: Verify that following are members of exponential family:
a) Geometric distribution p(x) = p²-¹(1 - p);
b) Poisson distribution p(x) = e-;
c) Normal distribution N(μ, 02);
d) Beta distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1be09dda-a703-4921-a1af-755c7a3c2844%2F1e3fe418-1be8-4267-b908-ffa7a4136ec7%2F73r6z8e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Instructions for Exercises 1 to 4: In each of these exercises, assume that the random variables
X₁, Xn form a random sample of size n form the distribution specified in that exercise,
and show that the statistic T specified in the exercise is a sufficient statistic for the parameter:
Exercise 1: A normal distribution for which the mean is known and the variance o² is
unknown; T = ²₁ (X₁-µ)².
Exercise 2: A gamma distribution with parameters a and 3, where the value of 3 is known
and the value of a is unknown (a > 0); T = II/1 X₁.
Exercise 3: A uniform distribution on the interval [a, b], where the value of a is known and
the value of b is unknown (b> a); T = max(X₁, Xn).
..
7
Exercise 4: A uniform distribution on the interval [a, b], where the value of b is known and
the value of a is unknown (b> a); T = min(X₁,, Xn).
... "
Exercise 5: Suppose that X₁, Xn form a random sample from a gamma distribution
with parameters a > 0 and 3 > 0, and the value of 3 is known. Show that the statistic
T = log X, is a sufficient statistic for the parameter a.
Exercise 6: The Pareto distribution has density function:
f(x|xo,0) = 0xx-0-1, x ≥
Σ
0, 0>1
Assume that ro> 0 is given and that X₁, X2,, Xn is an i.i.d. sample. Find a sufficient
statistic for by (a) using factorization theorem, (b) using the property of exponential family.
Are they the same? If not, why are both of them sufficient?
Exercise 7: Verify that following are members of exponential family:
a) Geometric distribution p(x) = p²-¹(1 - p);
b) Poisson distribution p(x) = e-;
c) Normal distribution N(μ, 02);
d) Beta distribution.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![MATLAB: An Introduction with Applications](https://www.bartleby.com/isbn_cover_images/9781119256830/9781119256830_smallCoverImage.gif)
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
![Probability and Statistics for Engineering and th…](https://www.bartleby.com/isbn_cover_images/9781305251809/9781305251809_smallCoverImage.gif)
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
![Statistics for The Behavioral Sciences (MindTap C…](https://www.bartleby.com/isbn_cover_images/9781305504912/9781305504912_smallCoverImage.gif)
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
![MATLAB: An Introduction with Applications](https://www.bartleby.com/isbn_cover_images/9781119256830/9781119256830_smallCoverImage.gif)
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
![Probability and Statistics for Engineering and th…](https://www.bartleby.com/isbn_cover_images/9781305251809/9781305251809_smallCoverImage.gif)
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
![Statistics for The Behavioral Sciences (MindTap C…](https://www.bartleby.com/isbn_cover_images/9781305504912/9781305504912_smallCoverImage.gif)
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
![Elementary Statistics: Picturing the World (7th E…](https://www.bartleby.com/isbn_cover_images/9780134683416/9780134683416_smallCoverImage.gif)
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
![The Basic Practice of Statistics](https://www.bartleby.com/isbn_cover_images/9781319042578/9781319042578_smallCoverImage.gif)
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
![Introduction to the Practice of Statistics](https://www.bartleby.com/isbn_cover_images/9781319013387/9781319013387_smallCoverImage.gif)
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman