Variance of an aggregate Poisson total

MediumRandom Variables Stochastic Processes~4m

Claims arrive as a Poisson process with mean $\lambda = 4$ per day. Each claim's size is independent and equally likely to be $1, $2, or $3. Let $S$ be the total claim amount in a day. What is the variance of $S$ ? Give a decimal to four places.

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Compound Poisson: use the law of total variance.
$\operatorname{Var}(S)=\lambda\,E[X^2]$ with $E[X^2]=14/3$ .

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18.6667 (± 0.001)

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Variance of an aggregate Poisson total

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