It is known that the probability distribution given a random variable Y with the parameter θ belongs to the Exponential Family of distributions if we can write f(y;θ) as the following in the picture attached. Given the picture attached, answer all parts a, b, and c of the question. Show all work and all steps. f(y;0) = exp[a(y)-b(0) + c(0) + d(y)](Eq. 1)Suppose binary random variable X~Bernoulli(t) with P(X=1)=Dn and P(X=0)=1-1.a) Show that this probability distribution belongs to the exponential family of distributions,b) Provide the form of the functions a, b, c, and d (Eq.1) for the Bernoulli distribution. Isthis distribution in the canonical form? What is the natural parameter of thisdistribution?c) Using the properties of distributions in the exponential family calculate:i.E(X)ii.Var(X).

Given information: The probability density function is as given below: fy;θ=eay.bθ+cθ+dyX~Bernoulliπ

Suppose binary random variable X~Bernoulli(r) with P(X=1)=n and P(X=0)=1-rt. a) Show that this probability distribution belongs to the exponential family of distributions, b) Provide the form of the functions a, b, c, and d (Eq.1) for the Bernoulli distribution. Is this distribution in the canonical form? What is the natural parameter of this distribution? c) Using the properties of distributions in the exponential family calculate: i. E(X)

Suppose binary random variable X~Bernoulli(r) with P(X=1)=n and P(X=0)=1-rt. a) Show that this probability distribution belongs to the exponential family of distributions, b) Provide the form of the functions a, b, c, and d (Eq.1) for the Bernoulli distribution. Is this distribution in the canonical form? What is the natural parameter of this distribution? c) Using the properties of distributions in the exponential family calculate: i. E(X)

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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Question

It is known that the probability distribution given a random variable Y with the parameter θ belongs to the Exponential Family of distributions if we can write f(y;θ) as the following in the picture attached.

Given the picture attached, answer all parts a, b, and c of the question. Show all work and all steps.

f(y;0) = exp[a(y)-b(0) + c(0) + d(y)]
(Eq. 1)
Suppose binary random variable X~Bernoulli(t) with P(X=1)=Dn and P(X=0)=1-1.
a) Show that this probability distribution belongs to the exponential family of distributions,
b) Provide the form of the functions a, b, c, and d (Eq.1) for the Bernoulli distribution. Is
this distribution in the canonical form? What is the natural parameter of this
distribution?
c) Using the properties of distributions in the exponential family calculate:
i.
E(X)
ii.
Var(X).

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