Let X be a random variable with probability mass function P(X = 1) = 1 2 , P(X = 2) = 1 3 , P(X = 5) = 1 6 (a) Find a function g such that E[g(X)] = 1 3 ln(2) + 1 6 ln(5). Your answer should give at least the values g(k) for all possible values k of X, but you can also specify g on a larger set if possible. (b) Let t be some real numb

The question is incomplete. The complete question is :

Let X be a random variable with probability mass function

P(X =1) =1/2, P(X=2)=1/3, P(X=5)=1/6

(a) Find a function g such that E[g(X)]=1/3 ln(2) + 1/6 ln(5). You answer should give at least the values g(k) for all possible values of k of X, but you can also specify g on a larger set if possible.

(b) Let t be some real number. Find a function g such that E[g(X)] =1/2 e^t + 2/3 e^(2t) + 5/6 e^(5t)

Solution :

Given :

a). We know :

So,  

Therefore comparing both the sides,

Also,  

b).

We known that

∴

Therefore on comparing, we get

∴

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .