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.

Short answer

Utilizing the linearity of expectation and the formula for the mean of the Poisson distribution, we hold that the needed number is$1.5.$

Step-by-step solution

Step 1:Given information

Given in the question that 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 $.7.$

Step 2:Explanation

Define random variables $X,Y,Z$that marks the number of accidents on these highways, respectively. We are given that

$X~Pois(0.3)$

$Y~Pois(0.5)$

$Z~Pois(0.7)$

Observe that the total number of accidents (call it $N$) can be written as $N=X+Y+Z.$Using the linearity of expectation and the formula for mean of Poisson distribution, we have that

localid="1646902573928" $E\left(N\right)=E\left(X\right)+E\left(Y\right)+E\left(Z\right)=0.3+0.5+0.7=1.5$

Step 3:Final answer

Utilizing the linearity of expectation and the formula for the mean of the Poisson distribution, we hold that the needed number is$1.5.$