In good years, quarrels between Tolstoy and his wife occurred according to a Poisson process with rate λ = 5 per month. In bad years, it was a Poisson process with rate μ = 10 per month. Suppose each year was equally likely to be a good or a bad year independently of what happened in previous years.
(a) Compute the mean and variance of the total number of quarrels in a randomly selected year.
(b) Suppose there were a total of 12 quarrels in January 1891. Compute the conditional probability mass function of the total number of quarrels in the year 1891 given this information. Hint: was this a good year or a bad year?
Here's my approach so far:
G = next year is a good year B = next year is a bad year N(t) = number of quarrels before time (month) t
$P(G) = P(B) = 0.5$
$N(t)|G \sim Pois(5t)$
$N(t)|B \sim Pois(10t)$
Is it correct to say:
$$P(N(t) = n) = P(N(t) = n | G)P(G) + P(N(t) = n | B)P(B)$$
$$P(N(t) = n) = \dfrac{(5t)^ne^{-5t}}{n!}0.5 + \dfrac{(10t)^ne^{-10t}}{n!}0.5$$
$$E[N(t=12)] = \sum_{i=0}^{\infty}n[\dfrac{(5t)^ne^{-5t}}{n!}0.5 + \dfrac{(10t)^ne^{-10t}}{n!}0.5]$$
and obtaining the variance would be a similar process?