Three Different Roads Feed into a Particular Freeway Entrance A What Is the Expected Total Number of Cars Entering

Essay

Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the table. $$\begin{array} { l l l l } \hline & \text { Road 1 } & \text { Road 2 } & \text { Road 3 } \\\hline \text { Expected value } & 750 & 1000 & 550 \\\\& 16 & 24 & 18 \\\text { Standard deviation } & & & \\\hline\end{array}$$
a. What is the expected total number of cars entering the freeway at this point during the period? (Hint: Let $X _ { i } , = \text { the number from road } i \text {.) }$
b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the numbers of cars on the different roads?
c. With $\bar{X} _ { i}$
denoting the number of cars entering from road I during the period, suppose that $\operatorname { Cov } \left( X _ { 1 } , \quad X _ { 2 } \right) = 80 , \quad \operatorname { Cov } \left( X _ { 1 } , \quad X _ { 3 } \right) = 90 \text {, and } \operatorname { Cov } \left( X _ { 2 } , X _ { 3 } \right) = 100$
(so that the three streams of traffic are not independent). Compute the expected total number of entering cars and the standard deviation of the total.