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Chapter 5: Q69E (page 242)

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.

a. What is the expected total number of cars entering the freeway at this point during the period? (Hint: Let \({\rm{Xi = }}\)the number from road\({\rm{i}}\).)

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 \({\rm{Xi}}\) denoting the number of cars entering from road\({\rm{i}}\)during the period, suppose that \({\rm{cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = 80\; and\; cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 90\; and\; cov}}\left( {{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 100}}\) (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.

Short Answer

Expert verified

a) \({\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 2400}}\)

b) \({\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 1205}}\)

c) \({\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 2400;\sigma = 41}}{\rm{.77}}\)

Step by step solution

01

Definition of standard deviation

The square root of the variance is the standard deviation of a random variable, sample, statistical population, data collection, or probability distribution. It is less resilient in practice than the average absolute deviation, but it is algebraically easier.

02

Determining expected total number of cars entering the freeway at this point during the period

Let \({{\rm{X}}_{\rm{i}}}\)be the number from road \({\rm{i}}\)as a clue.

The anticipated figure is

\(\begin{array}{*{20}{c}}{{\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = E}}\left( {{{\rm{X}}_{\rm{1}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{2}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{3}}}} \right)}\\{{\rm{ = 800 + 1000 + 600 = 2400}}}\end{array}\)

03

Determining the variance of the total number of entering cars

The variance can only be calculated if we assume independence.

\({{\rm{X}}_{\rm{i}}}\)is a random variable. As a result of assuming independence, the following is true:

\(\begin{array}{*{20}{c}}{{\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right)\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}} \right){\rm{ + V}}\left( {{{\rm{X}}_{\rm{2}}}} \right){\rm{ + V}}\left( {{{\rm{X}}_{\rm{3}}}} \right)}\\{{\rm{ = 1}}{{\rm{6}}^{\rm{2}}}{\rm{ + 2}}{{\rm{5}}^{\rm{2}}}{\rm{ + 1}}{{\rm{8}}^{\rm{2}}}}\\{{\rm{ = 1205}}}\end{array}\)

(1) Because of its autonomy.

04

Determining the expected total number of entering cars and the standard deviation of the total.

The anticipated figure is

\(\begin{array}{*{20}{c}}{{\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = E}}\left( {{{\rm{X}}_{\rm{1}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{2}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{3}}}} \right)}\\{{\rm{ = 800 + 1000 + 600 = 2400}}}\end{array}\)

Nothing appears to have changed. The cumulative expectation is unaffected by independence or dependence. The variance, however, will alter. The following is true:

\(\begin{array}{*{20}{c}}{{\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}} \right)\mathop {\rm{ = }}\limits^{{\rm{(2)}}} {\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}} \right){\rm{ + V}}\left( {{{\rm{X}}_{\rm{2}}}} \right){\rm{ + V}}\left( {{{\rm{X}}_{\rm{3}}}} \right)}\\{{\rm{Xi 5}}}\\{{\rm{ = 1}}{{\rm{6}}^{\rm{2}}}{\rm{ + 2}}{{\rm{5}}^{\rm{2}}}{\rm{ + 1}}{{\rm{8}}^{\rm{2}}}{\rm{ + 2 \times 80 + 2 \times 90 + 2 \times 100}}}\\{{\rm{ = 1745}}}\end{array}\)

(2): For any three random variables \({{\rm{X}}_{\rm{i}}}\), this equivalence holds.

The standard deviation of the data is

\({\rm{\sigma = }}\sqrt {{\rm{1745}}} {\rm{ = 41}}{\rm{.77}}\)

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