The random variable X has the N(μ,theta^2) distribution.
a) Show the moment generating fumction-have done this.
b) Define Y = e^−X and deduce from part (a) the values of E[Y ],
E[Y^2] and hence var(Y ).
c) Give an expression for E[XY ] in terms of MX(t) and hence calculate E[XY ].
d)Show that the covariance of X and Y can be written in the form
Cov(X, Y ) = -theta^2exp(theta^2/(2-μ) and derive an expression for the correlation.
have done part a.
Have done part b i think, since the moment generating function is
E[exp(tx)] let t=-1. then E[exp(-x)]=E[y]. Same foe t=-2 for E[y^2], then subtract for the variance.
Little stuck here and on the next section, we want E[xy], in terms of the moment generating function and then calculate.
The covariance is given by: E[xy]-E[x]E[y], hence from the part before this should make more sense, correlation, will be generated from what we have found previous,
Just really struggling on part c, the rest will follow from that,
any help would be most appreciated, many thanks.