In a good year, an organism births $\lambda_1$ offspring per month. In a bad year, the organism births $\lambda_2$ offspring. Of course, $\lambda_1 > \lambda_2$. If we suppose each year was equally and independently like to be a good or bad year, how do we approach:
i) Computing the mean and variance of a randomly selected year?
ii) Computing the conditional pmf of the offspring in a year if given the info that $n$ offspring have been born that year?
$X$ is the number of offsprings
$E[X]=E[X|\text{good year}]P(\text{good year})+E[X|\text{bad year}]P(\text{bad year})=\frac{\lambda_1+\lambda_2}{2}$
$E[X^2]=E[X^2|\text{good year}]P(\text{good year})+E[X^2|\text{bad year}]P(\text{bad year})=\frac{\lambda_1^2+\lambda_1+\lambda_2^2+\lambda_2}{2}$
$Var(X)=E[X^2]-E[X]^2=...$
For the second part I assume the question asks for the pmf of a year being a good year/bad year (otherwise it is trivial). Using Bayes formula
$P(\text{good year}|X=n)=\frac{P(X=n|\text{good year})P(\text{good year})}{P(X=n)}=\frac{P(X=n|\text{good year})P(\text{good year})}{P(X=n|\text{good year})P(\text{good year})+P(X=n|\text{bad year})P(\text{bad year})}=...$