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10 Generalized Linear Regression In the problems of this section $x^T \beta = \beta_0 + \sum_{i=1}^p \beta_i x_i$ Problem 10.1. $\epsilon \sim \text{Logistic}(0, 1)$, if its probability density function (pdf) is $\sigma'(x) = \frac{d}{dx}\sigma(x) = \frac{e^{-x}}{(1 + e^{-x})^2}$ a) Verify the primitive function $\int \sigma'(t)dt = \frac{1}{1 + e^{-x}} + C$ and determine C. Set $\sigma(x) := \frac{1}{1 + e^{-x}}$. b) Show that $P(-\epsilon \le x) = P(\epsilon \le x)$. You may use (131) even if you have failed to show it. c) Define $Y^* = x^T \beta + \epsilon$ and construct $Y = \begin{cases} 1 & \text{if } Y^* > 0\\ 0 & \text{otherwise.} \end{cases}$ Verify that $P(Y = 1 | x) = \sigma(x^T \beta)$. You may use (132) even if you have failed to show it. (133) d) The r.v. Y in this assignment is a special case of generalized linear regression (GLM). The way to construct a GLM goes via a link function. Present and justify the link function in this assignment.

          10 Generalized Linear Regression
In the problems of this section
$x^T \beta = \beta_0 + \sum_{i=1}^p \beta_i x_i$
Problem 10.1.
$\epsilon \sim \text{Logistic}(0, 1)$, if its probability density function (pdf) is
$\sigma'(x) = \frac{d}{dx}\sigma(x) = \frac{e^{-x}}{(1 + e^{-x})^2}$
a) Verify the primitive function
$\int \sigma'(t)dt = \frac{1}{1 + e^{-x}} + C$
and determine C. Set
$\sigma(x) := \frac{1}{1 + e^{-x}}$.
b) Show that
$P(-\epsilon \le x) = P(\epsilon \le x)$.
You may use (131) even if you have failed to show it.
c) Define
$Y^* = x^T \beta + \epsilon$
and construct
$Y = \begin{cases} 1 & \text{if } Y^* > 0\\ 0 & \text{otherwise.} \end{cases}$
Verify that
$P(Y = 1 | x) = \sigma(x^T \beta)$.
You may use (132) even if you have failed to show it.
(133)
d) The r.v. Y in this assignment is a special case of generalized linear regression (GLM). The
way to construct a GLM goes via a link function. Present and justify the link function in
this assignment.

10 Generalized Linear Regression
In the problems of this section
x^T β = β0 + ∑i=1^p xi
Problem 10.1.
ϵ∼Logistic(0, 1), if its probability density function (pdf) is
σ'(x) = (d)/(dx)σ(x) = (e^-x)/((1 + e^-x)^2)
a) Verify the primitive function
∫σ'(t)dt = (1)/(1 + e^-x) + C
and determine C. Set
σ(x) := (1)/(1 + e^-x).
b) Show that
P(-ϵ≤ x) = P(ϵ≤ x).
You may use (131) even if you have failed to show it.
c) Define
Y^* = x^T β + ϵ
and construct
Y = 1 if Y^* > 0
0 otherwise.
Verify that
P(Y = 1 | x) = σ(x^T β).
You may use (132) even if you have failed to show it.
(133)
d) The r.v. Y in this assignment is a special case of generalized linear regression (GLM). The
way to construct a GLM goes via a link function. Present and justify the link function in
this assignment.

Added by Keith R.

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