Please answer all parts a, b, and c of this question. Show all work.

A probability distribution of a random variable Y with parameters belongs to the Exponential Family of distributions if we can write f(y;) as: f(y;) = exp[(y)*b() + c() + d(y)] (Eq. 1)

Suppose binary random variable X~Bernoulli() with P(X=1)= and P(X=0)=1-.

a) Show that this probability distribution belongs to the exponential family of distributions, following the notation of your textbook (chapter 3).

b) Provide the form of the functions , 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).