Question

There are three highways in the county. the number of daily accidents that occur on these highways are poisson random variables with respective parameters .3, .5, and .7. find the expected number of accidents that will happen on any of these highways today.

Answer 1

There is a theorem that we can use to find the expected value of a random variable that is a sum of other random variables as follows.

If $X=X_1 + X_2 + ...+X_k$, then $E(X)=E(X_1)+E(X_2)+...+E(X_k)$.

In this case, let X = the number of accidents that will happen on any of those highways today,
$X_1,X_2,X_3$ are the numbers of accidents on each highway, respectively.

Then  $X = X_1+X_2+X_3$.

Since $X_1,X_2,X_3$ are Poisson variates, their expected values are the parameters given, .3, .5 and .7.

So $E(X_1)=.3, E(X_2) = .5, E(X_3) = .7$
Thus, $E(X)=E(X_1)+E(X_2)+E(X_3) = .3 + .5 + .7 = 1.5$

The answer is 1.5.