Q.57 - A First Course in Probability

Question

Suppose that the expected number of accidents per week at an industrial plant is $5$. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of $2.5$. If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week .

Step-by-Step Solution

Answer

$E\left(NX\right)=12.5$

Step 1: Given information

Given in the question that, . Suppose that the expected number of accidents per week at an industrial plant is $5$. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of $2.5$.

Step 2: Explanation

Let $N$ = number of accidents to happen in seven days. Allow $X=$ to number of workers injured in every mishap. The quantity of workers injured in accidents each week should be $N\times X$. So we needs $E[N X]$

Since we know that $\mathrm{N}$ and $\mathrm{X}$ are independent, we know that

$Cov(N,X)=E\left(NX\right)-E\left(N\right)E\left(X\right)=0$

$\Rightarrow E\left(NX\right)=E\left(N\right)E\left(X\right)=5\times 2.5=12.5$

Step 3: Final answer

The expected number of workers injured in a week is$E\left(NX\right)=12.5$