In Exercises 1-4, X₁, X₂, , Xn are i.i.d. experiments of a random variable X and X1, X2,,n are the sample outcomes, which we often abbreviate by the shorthand no- tation , and is the sample mean. Exercise 1 Let X~ Uniform([a, b]). Let â and 6 be the MLE estimators of a and b, respectively. (i) Show that and use this to show that ã = â E(ô - â) - = 1 1 ¹ ₁ (b − a), b = b + ₁₂ ² ₁ (b − a) – â), (ô n 1 n 1 - are unbiased estimators of a and b, respectively. (ii) Compute the MLE estimates a, b of a, b for the following sample data ☎¹: n 1 (b − a) n+ 1 - -2.45 0.65 -0.35 -2.38 -1.26 0.58 -1.52 -1.33 -0.42 -1.39 0.01 -1.87 -0.67 -0.34 -0.33 -1.45 0.02 (iii) Compute the unbiased estimates a, b of a, b for the same data above. ã,
In Exercises 1-4, X₁, X₂, , Xn are i.i.d. experiments of a random variable X and X1, X2,,n are the sample outcomes, which we often abbreviate by the shorthand no- tation , and is the sample mean. Exercise 1 Let X~ Uniform([a, b]). Let â and 6 be the MLE estimators of a and b, respectively. (i) Show that and use this to show that ã = â E(ô - â) - = 1 1 ¹ ₁ (b − a), b = b + ₁₂ ² ₁ (b − a) – â), (ô n 1 n 1 - are unbiased estimators of a and b, respectively. (ii) Compute the MLE estimates a, b of a, b for the following sample data ☎¹: n 1 (b − a) n+ 1 - -2.45 0.65 -0.35 -2.38 -1.26 0.58 -1.52 -1.33 -0.42 -1.39 0.01 -1.87 -0.67 -0.34 -0.33 -1.45 0.02 (iii) Compute the unbiased estimates a, b of a, b for the same data above. ã,
A First Course in Probability (10th Edition)
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Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![In Exercises 1-4, X₁, X₂, , Xn are i.i.d. experiments of a random variable X and
X1, X2,,n are the sample outcomes, which we often abbreviate by the shorthand no-
tation , and is the sample mean.
Exercise 1
Let X~ Uniform([a, b]). Let â and 6 be the MLE estimators of a and b, respectively.
(i) Show that
and use this to show that
ã =
â
E(ô - â)
-
=
1
1
¹ ₁ (b − a), b = b + ₁₂ ² ₁ (b − a)
– â),
(ô
n
1
n
1
-
are unbiased estimators of a and b, respectively.
(ii) Compute the MLE estimates a, b of a, b for the following sample data ☎¹:
n 1
(b − a)
n+ 1
-
-2.45 0.65 -0.35 -2.38 -1.26 0.58 -1.52 -1.33 -0.42
-1.39 0.01 -1.87 -0.67 -0.34 -0.33 -1.45 0.02
(iii) Compute the unbiased estimates a, b of a, b for the same data above.
ã,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33efa0ee-e3c4-4640-bf2a-d63d72536f00%2Fe6614b04-a63e-4056-be45-92c0f07cb028%2Fgu0chlj_processed.png&w=3840&q=75)
Transcribed Image Text:In Exercises 1-4, X₁, X₂, , Xn are i.i.d. experiments of a random variable X and
X1, X2,,n are the sample outcomes, which we often abbreviate by the shorthand no-
tation , and is the sample mean.
Exercise 1
Let X~ Uniform([a, b]). Let â and 6 be the MLE estimators of a and b, respectively.
(i) Show that
and use this to show that
ã =
â
E(ô - â)
-
=
1
1
¹ ₁ (b − a), b = b + ₁₂ ² ₁ (b − a)
– â),
(ô
n
1
n
1
-
are unbiased estimators of a and b, respectively.
(ii) Compute the MLE estimates a, b of a, b for the following sample data ☎¹:
n 1
(b − a)
n+ 1
-
-2.45 0.65 -0.35 -2.38 -1.26 0.58 -1.52 -1.33 -0.42
-1.39 0.01 -1.87 -0.67 -0.34 -0.33 -1.45 0.02
(iii) Compute the unbiased estimates a, b of a, b for the same data above.
ã,
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