Deriving Covariance between S and N: E(SN) - ìXìN

[johnnytzf]

consider the following model for aggregate claim amounts S:
S=X1+X2+...+XN
where the Xi are independent, identically distributed random
variables representing individual claim amounts and N is a random
variable,independent of the Xi and representing the number of
claims.let X has ìx and let N has mean ìN and variance ó²N.

a) show that E(SN)=ìX ( ì²N + ó²N ) by considering expected values
conditional on the value of N

b) hence derive an expression for the covariance between S and N.



I know that

E(S) = E(E(S|N)) = E(N)E(S) = ìXìN,
Var(E(S|N)) = Var(N)Var(S) = ó²Xó²N

but how to link it with E(SN)??

 

[Stephen Tashi]

Let's make the question more readable. (See post #3 in the thread https://www.physicsforums.com/showthread.php?t=617567)

Is it this?:

Consider the following model for aggregate claim amounts [itex] S [/itex]
[itex] S=X_1+X_2+...+X_N [/itex]
where the [itex] X_i [/itex] are independent, identically distributed random
variables representing individual claim amounts and [itex] N [/itex] is a random
variable,independent of the [itex] X_i [/itex] and representing the number of
claims. let the mean of [itex] X_i [/itex] be [itex] \mu_X [/itex] and let the mean of [itex] N [/itex] be [itex] \mu_N [/itex]. Let the variance of [itex] N [/itex] be [itex] \sigma^2_N [/itex]. .

a) show that [itex] E(NS)=\mu_X ( \mu^2_N + \sigma^2_N ) [/itex] by considering expected values
conditional on the value of [itex] N [/itex]

b) hence derive an expression for the covariance between [itex] S [/itex] and [itex]N [/itex].