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I am busy with ruin theory. S(t) = Sum of Xi's from i=1 to N(t)

$S(t)$ is the aggregate claim size after $t$ years, where $X_i$ is the individual claim size (with mean and variance given) and $N(t)$ is the number of claims that follow a Poisson distribution with parameter $\lambda$, and it is assumed that $\lambda$ has an exponential distribution with given mean.

Now since $N(t)$ has a Poisson distribution, $S(t)$ has a Compound Poisson distribution with parameter $\lambda$, right?

Then is the expected value of $S(t)$:

$E[S(t)] = E[N(t)]\cdot E[X_i]$
$\,\:\quad\qquad = (E[\lambda]\cdot t)\cdot E[X_i]\,$ ?

I am specifically confused about the $E[N(t)]$ part, does it include the $t$ variable even though it is only distributed Poisson($\lambda$) or not? And then do you use $E[\lambda]$ or only $\lambda$ in calculating the $E[N(t)]$?

Also, the variance of $N(t)$, is it equal to the variance of $\lambda$ or the expected value of $\lambda$? And again should it be multiplied by $t$ even though $t$ is not given as part of the parameter in the question?

Would be so glad if anyone can help. And please ask if there is any more information needed to answer the question.

I am busy with ruin theory.

$$ S(t) = \sum_{i=1}^{N(t)} X_i $$

$S(t)$ is the aggregate claim size after $t$ years, where $X_i$ is the individual claim size (with mean and variance given) and $N(t)$ is the number of claims that follow a Poisson distribution with parameter $\lambda$, and it is assumed that $\lambda$ has an exponential distribution with given mean.

Now since $N(t)$ has a Poisson distribution, $S(t)$ has a Compound Poisson distribution with parameter $\lambda$, right?

Then is the expected value of $S(t)$:

$E[S(t)] = E[N(t)]\cdot E[X_i]$
$\,\:\quad\qquad = (E[\lambda]\cdot t)\cdot E[X_i]\,$ ?

I am specifically confused about the $E[N(t)]$ part, does it include the $t$ variable even though it is only distributed Poisson($\lambda$) or not? And then do you use $E[\lambda]$ or only $\lambda$ in calculating the $E[N(t)]$?

Also, the variance of $N(t)$, is it equal to the variance of $\lambda$ or the expected value of $\lambda$? And again should it be multiplied by $t$ even though $t$ is not given as part of the parameter in the question?

Would be so glad if anyone can help. And please ask if there is any more information needed to answer the question.

I am busy with ruin theory.

$S(t)$ is the aggregate claim size after $t$ years, where $X_i$ is the individual claim size (with mean and variance given) and $N(t)$ is the number of claims that follow a Poisson distribution with parameter $\lambda$, and it is assumed that $\lambda$ has an exponential distribution with given mean.

Now since $N(t)$ has a Poisson distribution, $S(t)$ has a Compound Poisson distribution with parameter $\lambda$, right?

Then is the expected value of $S(t)$:

$E[S(t)] = E[N(t)]\cdot E[X_i]$
$\,\:\quad\qquad = (E[\lambda]\cdot t)\cdot E[X_i]\,$ ?

I am specifically confused about the $E[N(t)]$ part, does it include the $t$ variable even though it is only distributed Poisson($\lambda$) or not? And then do you use $E[\lambda]$ or only $\lambda$ in calculating the $E[N(t)]$?

Also, the variance of $N(t)$, is it equal to the variance of $\lambda$ or the expected value of $\lambda$? And again should it be multiplied by $t$ even though $t$ is not given as part of the parameter in the question?

Would be so glad if anyone can help. And please ask if there is any more information needed to answer the question.

I am busy with ruin theory. S(t) = Sum of Xi's from i=1 to N(t)

$S(t)$ is the aggregate claim size after $t$ years, where $X_i$ is the individual claim size (with mean and variance given) and $N(t)$ is the number of claims that follow a Poisson distribution with parameter $\lambda$, and it is assumed that $\lambda$ has an exponential distribution with given mean.

Now since $N(t)$ has a Poisson distribution, $S(t)$ has a Compound Poisson distribution with parameter $\lambda$, right?

Then is the expected value of $S(t)$:

$E[S(t)] = E[N(t)]\cdot E[X_i]$
$\,\:\quad\qquad = (E[\lambda]\cdot t)\cdot E[X_i]\,$ ?

I am specifically confused about the $E[N(t)]$ part, does it include the $t$ variable even though it is only distributed Poisson($\lambda$) or not? And then do you use $E[\lambda]$ or only $\lambda$ in calculating the $E[N(t)]$?

Also, the variance of $N(t)$, is it equal to the variance of $\lambda$ or the expected value of $\lambda$? And again should it be multiplied by $t$ even though $t$ is not given as part of the parameter in the question?

Would be so glad if anyone can help. And please ask if there is any more information needed to answer the question.