please tell all parts of question f (f)For each of the following families of distributions, explain whether it is an exponential family:Bernoulli(p), p E (0, 1)Ν (μ, 1), μeRΝ (μ, σ') , με R md σ? >0Exponential(A), 1> 0Uniform([0, O]), 0 >0• r(a, 3), where a, 3 > 0• Poisson(A), > 0• Binomial with parameters n and p• Binomial with n = 100 and other parameter being pCauchy distribution centered around 0.

Name: please tell all parts of question f (f)For each of the following famil
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Description: please tell all parts of question f (f)For each of the following families of distributions, explain whether it is an exponential family:Bernoulli(p), p E (0, 1)Ν (μ, 1), μeRΝ (μ, σ') , με R md σ? >0Exponential(A), 1> 0Uniform([0, O]), 0 >0• r(a, 3),

Answered: (f) For each of the following families…

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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please tell all parts of question f

(f)
For each of the following families of distributions, explain whether it is an exponential family:
Bernoulli(p), p E (0, 1)
Ν (μ, 1), μeR
Ν (μ, σ') , με R md σ? >0
Exponential(A), 1> 0
Uniform([0, O]), 0 >0
• r(a, 3), where a, 3 > 0
• Poisson(A), > 0
• Binomial with parameters n and p
• Binomial with n = 100 and other parameter being p
Cauchy distribution centered around 0.

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A k-parameter exponential family ${f (x; θ) : θ \in Ω \subseteq R^{k}}$ of distributions that can be expressed as $f (x; θ) = e x p [\sum_{i = 1}^{k} u_{i} (θ) T_{i} (x) + ν (θ) + w (x)]$ with the following regularity conditions:

$1 . T h e s u p p o r t (x : f (x; θ) > 0) d o e s n o t d e p e n d o n 2 . T h e p a r a m e t e r s p a c e Ω i s a n o p e n s u b s e t i n R^{k}, c o n t a i n i n g k d i m e n s i o n a l r e c \tan g l e 3 . {1, T_{1} (x), T_{2} (x), . . . ., T_{k} (x)} a n d {1, u_{1} (θ), u_{2} (θ), . . . ., u_{k} (θ)} a r e l i n e a r l y i n d e p e n d e n t$

; is defined as k- parameter exponential family.

a.)

$X ~ B e r n o u l l i (p) f (x; p) = p^{x} {(1 - p)}^{1 - x} = {[\frac{p}{1 - p}]}^{x} (1 - p) = e x p [x \ln [\frac{p}{1 - p}]] (1 - p) H e r e, k = 1, u_{1} (p) = \ln [\frac{p}{1 - p}] T_{1} (x) = x v (p) = \ln (1 - p) T h u s, X i s a o n e - p a r a m e t e r e x p o n e n t i a l f a m i l y$

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