# Application Problem: Random Sums of Random Variables and Correlation

I am trying to answer the following:

The number of traffic accidents per year at a given intersection follows a Poisson(10, 000)-distribution. The number of deaths per accident follows a Poisson(0.1)-distribution, and the number of casualties per accidents follows a Poisson(2)-distribution. The correlation coefficient between the number of casualties and the number of deaths per accidents is 0.5. Compute the expectation and variance of the total number of deaths and casualties during a year.

I know I am supposed to apply the concepts from sums of a random number of random variables. To solve for the expectation and variance of the total number of deaths and casualties: I should use the formulas $E[S_{N}] = E[N] · E [X]$, $Var[S_N] = E[N] · Var[X] + (E[X])^{2}· Var[N]$.

Here $X_1, X_2, . . .$ are i.i.d. random variables, $N$ is a nonnegative, integer-valued random variable that is independent of $X_1, X_2, . . .$, and $S_{N} = X_1 + X_2 + · · · + X_{N}$.

In this application problem $N$ is the number of accidents per year at the intersection, while X is the number of deaths and casualties. I know for a poisson distribution the expected value and variance are both equal to the parameter value. I am lost how to compute the expected value and variance (the $E[X]$ and $Var[X]$ terms) of the number of deaths and casualties given the correlation coefficient.

## 1 Answer

For $i=1,2,\ldots,\;$ let,

\begin{eqnarray*} D_i &=& \text{#Deaths in accident $i$} \\ C_i &=& \text{#Casualties in accident $i$} \\ X_i &=& D_i+C_i. \end{eqnarray*}

So we want to find $E(X_i)$ and $Var(X_i)$ to plug into your formulas (with $X=X_i$).

\begin{eqnarray*} E(X_i) &=& E(D_i+C_i) \\ &=& E(D_i)+E(C_i) \\ &=& 2.1 \end{eqnarray*}

\begin{eqnarray*} Var(X_i) &=& Var(D_i+C_i) \\ &=& Var(D_i) + Var(C_i) + 2Cov(D_i,C_i) \qquad\qquad\qquad\qquad\qquad\text{(1)} \\ &=& Var(D_i) + Var(C_i) + 2\rho(D_i,C_i)\sqrt{Var(D_i)Var(C_i)} \\ &=& 0.1 + 2 + 2\times 0.5\sqrt{0.1\times 2} \\ &=& 2.1 + \sqrt{0.2}. \end{eqnarray*}

Note: (1) Formula for variance of sum of r.v.'s.