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Chapter 5: Q86SE (page 244)

A student has a class that is supposed to end at\({\rm{9:00 A}}{\rm{.M}}\). and another that is supposed to begin at \({\rm{9:10 A}}{\rm{.M}}\).

Suppose the actual ending time of the \({\rm{9:00 A}}{\rm{.M}}\). class is a normally distributed \({\rm{rv}}\) \({{\rm{X}}_{\rm{1}}}\)with mean \({\rm{9:02 A}}{\rm{.M}}\)and standard deviation \({\rm{1}}{\rm{.5\;min}}\)and that the starting time of the next class is also a normally distributed \({\rm{rv}}\) \({{\rm{X}}_{\rm{2}}}\)with mean \({\rm{9:10}}\)and standard deviation \({\rm{1\;min}}{\rm{.}}\)Suppose also that the time necessary to get from one classroom to the other is a normally distributed \({\rm{rv}}\) \({{\rm{X}}_{\rm{3}}}\)with mean \({\rm{6\;min}}\) and standard deviation\({\rm{1\;min}}\). What is the probability that the student makes it to the second class before the lecture starts? (Assume independence of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\), which is reasonable if the student pays no attention to the finishing time of the first class.)

Short Answer

Expert verified

The probability is \(P\left( {{X_1} + {X_3} \le {X_2}} \right) = 0.8340\)that the student makes it to the second class before the lecture starts.

Step by step solution

01

Definition

The standard deviation is a measurement of a collection of values' variance or dispersion. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation suggests that the values are dispersed over a larger range.

02

Calculating the probability

Notice that the student will not be late if the total of the finishing time and the time required to go from one classroom to the next is less than or equal to the start time of the following class. This can be expressed as an event.

\({X_1} + {X_3} \le {X_2}.\)

Thus, the random variable of interest is

\({{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}\)

The linear combination follows the same distribution as the three random variables. It is simple to calculate the probability by standardizing the random variable \({{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}\). The random variable's mean value equals

\(\begin{array}{l}{\rm{E}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}} \right)\\{\rm{ = E}}\left( {{{\rm{X}}_{\rm{1}}}} \right){\rm{ + E}}\left( {{{\rm{X}}_{\rm{3}}}} \right){\rm{ - E}}\left( {{{\rm{X}}_{\rm{2}}}} \right)\\{\rm{ = 2 + 6 - 10}}\\{\rm{ = - 2,}}\end{array}\)

03

Calculating the probability 

the variance, because of the independence, is

\(\begin{array}{l}{\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{3}}}{\rm{ - }}{{\rm{X}}_{\rm{2}}}} \right)\\{\rm{ = V}}\left( {{{\rm{X}}_{\rm{1}}}} \right){\rm{ + V}}\left( {{{\rm{X}}_{\rm{3}}}} \right){\rm{ + ( - 1}}{{\rm{)}}^{\rm{2}}}{\rm{V}}\left( {{{\rm{X}}_{\rm{2}}}} \right)\\{\rm{ = 1}}{\rm{.}}{{\rm{5}}^{\rm{2}}}{\rm{ + }}{{\rm{1}}^{\rm{2}}}{\rm{ + }}{{\rm{1}}^{\rm{2}}}\\{\rm{ = 4}}{\rm{.25}}\end{array}\)

and the standard deviation is

\({\rm{\sigma = }}\sqrt {{\rm{4}}{\rm{.25}}} {\rm{ = 2}}{\rm{.06}}\)

Therefore, the probability of the mentioned event is

\(\begin{aligned}P\left( {{X_1} + {X_3}拢{X_2}} \right) &= P\left( {{X_1} + {X_3} - {X_2}拢0} \right) \\ &= P\left( {\frac{{{X_1} + {X_3} - {X_2} - E\left( {{X_1} + {X_3} - {X_2}} \right)}}{\sigma }拢\frac{{0 - ( - 2)}}{{2.06}}} \right) \\ &= P(Z拢0.97) \\ &= 0.8340 \\ \end{aligned} \)

(1): from the appendix's normal probability table Software can also be used to calculate the likelihood.

Therefore, the solution is \(P\left( {{X_1} + {X_3} \le {X_2}} \right) = 0.8340\).

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Most popular questions from this chapter

a. Use the general formula for the variance of a linear combination to write an expression for\({\rm{V(aX + Y)}}\). Then let\({\rm{a = }}{{\rm{\sigma }}_{\rm{\gamma }}}{\rm{/}}{{\rm{\sigma }}_{\rm{X}}}\), and show that. (Hint: Variance is always, and\({\rm{Cov(X,Y) = }}{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}{\rm{ \times \rho }}\).)

b. By considering\({\rm{V(aX - Y)}}\), conclude that\({\rm{\rho 拢 1}}\).

c. Use the fact that \({\rm{V(W) = 0}}\)only if \({\rm{W}}\)is a constant to show that \({\rm{\rho = 1}}\)only if\({\rm{Y = aX + b}}\).

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.

Answer the following questions:

a. Given that\({\rm{X = 1}}\), determine the conditional pmf of \({\rm{Y}}\)-i.e., \({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(0}}\mid {\rm{1),}}{{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(1}}\mid {\rm{1)}}\), and\({{\rm{p}}_{{\rm{Y}}\mid {\rm{X}}}}{\rm{(2}}\mid {\rm{1)}}\).

b. Given that two houses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island?

c. Use the result of part (b) to calculate the conditional probability\({\rm{P(Y拢

1}}\mid {\rm{X = 2)}}\).

d. Given that two houses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island?

There are two traffic lights on a commuter's route to and from work. Let \({{\rm{X}}_{\rm{1}}}\) be the number of lights at which the commuter must stop on his way to work, and \({{\rm{X}}_{\rm{2}}}\) be the number of lights at which he must stop when returning from work. Suppose these two variables are independent, each with pmf given in the accompanying table (so \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\) is a random sample of size \({\rm{n = 2}}\)).

a. Determine the pmf of \({{\rm{T}}_{\rm{o}}}{\rm{ = }}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{X}}_{\rm{2}}}\).

b. Calculate \({{\rm{\mu }}_{{{\rm{T}}_{\rm{o}}}}}\). How does it relate to \({\rm{\mu }}\), the population mean?

c. Calculate \({\rm{\sigma }}_{{{\rm{T}}_{\rm{o}}}}^{\rm{2}}\). How does it relate to \({{\rm{\sigma }}^{\rm{2}}}\), the population variance?

d. Let \({{\rm{X}}_{\rm{3}}}\) and \({{\rm{X}}_{\rm{4}}}\) be the number of lights at which a stop is required when driving to and from work on a second day assumed independent of the first day. With \({{\rm{T}}_{\rm{o}}}{\rm{ = }}\) the sum of all four \({{\rm{X}}_{\rm{i}}}\) 's, what now are the values of \({\rm{E}}\left( {{{\rm{T}}_{\rm{a}}}} \right)\) and \({\rm{V}}\left( {{{\rm{T}}_{\rm{a}}}} \right)\)?

e. Referring back to (d), what are the values of \({\rm{P}}\left( {{{\rm{T}}_{\rm{o}}}{\rm{ = 8}}} \right)\) and \(\text{P}\left( {{\text{T}}_{\text{e}}}\text{ }\!\!{}^\text{3}\!\!\text{ 7} \right)\) (Hint: Don't even think of listing all possible outcomes!)

a. Compute the covariance for \({\rm{X}}\) and \({\rm{Y}}\).

b. Compute \({\rm{\rho }}\) for \({\rm{X}}\) and \({\rm{Y}}\).
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