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(a) Write down the PDF of Y;. This is the marginal distribution of the ith component of the random vector Y = (Y₁,..., Yn). (b) Since the {} are independent, so are the {Y}. Using this fact, determine the PDF of the random vector Y (the joint PDF of Y₁,..., Yn). (c) Viewed as a function of o², ßo, and ₁ with the values of the {Y;} held fixed, the joint PDF from Part (b) is the likelihood function connected with this model. Show that for any fixed o² > 0, this function is maximized by those ß₁ and 3₁ which minimize the expression 12 Q(Bo, B₁)=(Y₁-Bo-B₁x;)². Ĺo i=1 Thus, the Band ₁ that minimize the function Q(Bo, B₁) above are the MLEs of the parameters ßo and ß₁ in this model. In applied statistics, this set up is known as a simple linear regression model and the Bo, 3₁ are called the least squares estimators of Bo and 3₁.

(a) Write down the PDF of Y;. This is the marginal distribution of the ith component of the random vector Y = (Y₁,..., Yn). (b) Since the {} are independent, so are the {Y}. Using this fact, determine the PDF of the random vector Y (the joint PDF of Y₁,..., Yn). (c) Viewed as a function of o², ßo, and ₁ with the values of the {Y;} held fixed, the joint PDF from Part (b) is the likelihood function connected with this model. Show that for any fixed o² > 0, this function is maximized by those ß₁ and 3₁ which minimize the expression 12 Q(Bo, B₁)=(Y₁-Bo-B₁x;)². Ĺo i=1 Thus, the Band ₁ that minimize the function Q(Bo, B₁) above are the MLEs of the parameters ßo and ß₁ in this model. In applied statistics, this set up is known as a simple linear regression model and the Bo, 3₁ are called the least squares estimators of Bo and 3₁.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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6. Let Y₁,..., Yn be (independent) random variables with the following structure: Y = Bo+ Biểi tê, i=1,..,”, where Bo, 3₁ are two unknown (real) numbers, the {x;} are given con- stants, and the {i} are IID normal variables with mean zero and an unknown variance o² > 0. (a) Write down the PDF of Y₁. This is the marginal distribution of the ith component of the random vector Y = (Y₁, ..., Yn). (b) Since the {} are independent, so are the {Y}. Using this fact, determine the PDF of the random vector Y (the joint PDF of Y₁,..., Yn). (c) Viewed as a function of o², Bo, and ₁ with the values of the {Y} held fixed, the joint PDF from Part (b) is the likelihood function connected with this model. Show that for any fixed o² > 0, this function is maximized by those ßo and 3₁ which minimize the expression n Q(Bo, B₁) = (Yi - Bo - B₁x;)². i=1 Thus, the Bo and ₁ that minimize the function Q(Bo, B₁) above are the MLEs of the parameters Bo and ₁ in this model. In applied statistics, this set up is known as a simple linear regression model and the Bo, 31 are called the least squares estimators of Bo and 3₁.
Transcribed Image Text:6. Let Y₁,..., Yn be (independent) random variables with the following structure: Y = Bo+ Biểi tê, i=1,..,”, where Bo, 3₁ are two unknown (real) numbers, the {x;} are given con- stants, and the {i} are IID normal variables with mean zero and an unknown variance o² > 0. (a) Write down the PDF of Y₁. This is the marginal distribution of the ith component of the random vector Y = (Y₁, ..., Yn). (b) Since the {} are independent, so are the {Y}. Using this fact, determine the PDF of the random vector Y (the joint PDF of Y₁,..., Yn). (c) Viewed as a function of o², Bo, and ₁ with the values of the {Y} held fixed, the joint PDF from Part (b) is the likelihood function connected with this model. Show that for any fixed o² > 0, this function is maximized by those ßo and 3₁ which minimize the expression n Q(Bo, B₁) = (Yi - Bo - B₁x;)². i=1 Thus, the Bo and ₁ that minimize the function Q(Bo, B₁) above are the MLEs of the parameters Bo and ₁ in this model. In applied statistics, this set up is known as a simple linear regression model and the Bo, 31 are called the least squares estimators of Bo and 3₁.
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