Question

8. $S_1$ and $S_2$ are random variables representing total claim amounts in two portfolios, and both can be well modeled by compound Poisson distributions. $S_1 = Y_1 + dots + Y_{N_1}$ where $N_1$ is Poisson with parameter $lambda_1 = 2$ and the claim size random variable Y has a distribution given by $Y = egin{cases} 200 ext{prob} = 0.5\ 300 ext{prob} = 0.3\ 400 ext{prob} = 0.2. end{cases}$ Similarly, $S_2 = Z_1 + dots + Z_{N_2}$ where $N_2$ is Poisson with parameter $lambda_2 = 3$ and the claim size random variable Y has a distribution given by $Z = egin{cases} 300 ext{prob} = 0.1\ 400 ext{prob} = 0.3\ 500 ext{prob} = 0.6. end{cases}$ If $S_1$ and $S_2$ are independent, what is the probability distribution of $S = S_1 + S_2$? What are its mean, variance and moment generating function? What is $P[S le 400]$? Use the recursive formula to find $P[S = r]$ for $r le 2000.$

          8. $S_1$ and $S_2$ are random variables representing total claim amounts in
two portfolios, and both can be well modeled by compound Poisson
distributions. $S_1 = Y_1 + dots + Y_{N_1}$ where $N_1$ is Poisson with parameter
$lambda_1 = 2$ and the claim size random variable Y has a distribution given
by
$Y = egin{cases} 200 	ext{prob} = 0.5\ 300 	ext{prob} = 0.3\ 400 	ext{prob} = 0.2. end{cases}$
Similarly, $S_2 = Z_1 + dots + Z_{N_2}$ where $N_2$ is Poisson with parameter
$lambda_2 = 3$ and the claim size random variable Y has a distribution given
by
$Z = egin{cases} 300 	ext{prob} = 0.1\ 400 	ext{prob} = 0.3\ 500 	ext{prob} = 0.6. end{cases}$
If $S_1$ and $S_2$ are independent, what is the probability distribution of
$S = S_1 + S_2$? What are its mean, variance and moment generating
function? What is $P[S le 400]$? Use the recursive formula to find
$P[S = r]$ for $r le 2000.$

8. S1 and S2 are random variables representing total claim amounts in
two portfolios, and both can be well modeled by compound Poisson
distributions. S1 = Y1 + dots + YN1 where N1 is Poisson with parameter
lambda1 = 2 and the claim size random variable Y has a distribution given
by
Y = egincases 200 extprob = 0.5 300 extprob = 0.3 400 extprob = 0.2. endcases
Similarly, S2 = Z1 + dots + ZN2 where N2 is Poisson with parameter
lambda2 = 3 and the claim size random variable Y has a distribution given
by
Z = egincases 300 extprob = 0.1 400 extprob = 0.3 500 extprob = 0.6. endcases
If S1 and S2 are independent, what is the probability distribution of
S = S1 + S2? What are its mean, variance and moment generating
function? What is P[S le 400]? Use the recursive formula to find
P[S = r] for r le 2000.

Added by Antonia M.

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