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).
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 31CR
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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.
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