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Chapter 5: Q1E (page 210)

A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let \({\rm{X}}\)denote the number of hoses being used on the self-service island at a particular time, and let\({\rm{Y}}\)denote the number of hoses on the full-service island in use at that time. The joint \({\rm{pmf}}\) of \({\rm{X}}\)and \({\rm{Y}}\) appears in the accompanying tabulation.

a. What is\({\rm{P(X = 1 and Y = 1)}}\)?

b. Compute P(X拢1}and{Y拢1)

c. Give a word description of the event , and compute the probability of this event.

d. Compute the marginal \({\rm{pmf}}\) of \({\rm{X}}\)and of \({\rm{Y}}\). Using \({{\rm{p}}_{\rm{X}}}{\rm{(x)}}\)what is P(X拢1)?

e. Are \({\rm{X}}\)and\({\rm{Y}}\)independent \({\rm{rv's}}\)? Explain

Short Answer

Expert verified

a) \({\rm{P(X = 1 and Y = 1) = 0}}{\rm{.2;}}\)

b) P(X拢1}and{Y拢1) = 0.42;

c) P(X0}and{Y0) = 0.7;

d)

\begin{aligned}{p_X}(0) &= 0.16,{p_X}(1) = 0.34 \hfill \\{p_X}(2) &= 0.5, \hfill \\{p_X}(x) &= 0,\;\;\;x"I \{ 0,1,2\} ; \hfill \\\begin{array}{}{{p_Y}(0) = 0.24,} \\{{p_Y}(1) = 0.38,} \\{{p_Y}(2) = 0.38,} \\{{p_Y}(y) = 0,\;\;\;y"I \{ 0,1,2\} } \\ \begin{gathered}P(X拢1) = 0.5; \hfill \\\; \hfill \\\end{gathered} \end{array} \hfill \\ \end{aligned}

e) dependent

Step by step solution

01

Definition of probability

The proportion of the total number of conceivable outcomes to the number of options in an exhaustive collection of equally likely outcomes that cause a given occurrence.

02

Determining the \({\rm{P(X = 1 and Y = 1)}}\)

Consider the discrete random variables \({\rm{X and Y}}\)

\({\rm{p(x,y)}}\)is the joint probability mass function

\({\rm{p(x,y) = P(X = x,Y = y)}}\)

and \(\mathop {\rm{{aa}}}\limits_{\rm{x}} {\rm{n}}\mathop {\rm{{aa}}}\limits_{\rm{y}} {\rm{np(x,y) = 1}}\)for every pair \({\rm{(x,y)}}\)

(a) The following is true, as we can see from the definition above.

\({\rm{P(X = 1\;and\;Y = 1) = p(1,1) = 0}}{\rm{.2}}\)

where the value can be found in the supplied table

03

Determining \({\rm{P(X£ 1 and Y£ 1)}}\)

(b) The following vacations

\(\begin{aligned}{}{P(X拢1\;and\;Y拢1)\mathop = \limits^{(1)} p(0,0) + p(0,1) + p(1,0) + p(1,1)} \\{= 0.1 + 0.04 + 0.08 + 0.2} \\{ = 0.42.}\end{aligned}\)

(1): the occurrence

\(\{ X拢1\;and\;Y拢1\} \)

can be written as the sum of four separate events

\({\rm{\{ X = 0,Y = 0\} ,\{ X = 0,Y = 1\} ,\{ X = 1,Y = 0\} ,\{ X = 1,Y = 1\} }}{\rm{.}}\)

04

Determining the probability of this event. 

(c) Because neither random variable is equal to zero, at least one of the hoses is used on both islands.

The likelihood of the given event is as follows:

\(\begin{aligned}{P(X0\;and\;Y0)\mathop= \limits^{(2)} p(1,1) + p(1,2) + p(2,1) + p(2,2)} \\{ = 0.2 + 0.06 + 0.14 + 0.3}\\{ = 0.7.}\end{aligned}\)

05

Determining the marginal \({\rm{pmf}}\) of \({\rm{X}}\) and of \({\rm{Y}}\)

(d) The mass function of marginal probability

random variable that is discrete \({\rm{X}}\) is

\({{\rm{p}}_{\rm{X}}}{\rm{(x) = }}\mathop {\rm{{aa}}}\limits_{\rm{y}} {\rm{np(x,y),}}\;\;\;{\rm{\;for every\;x}}\)

The mass function of marginal probability

random variable that is discrete \({\rm{Y}}\)is

\({{\rm{p}}_{\rm{Y}}}{\rm{(y) = }}\mathop {\rm{{aa}}}\limits_{\rm{x}} {\rm{np(x,y),}}\;\;\;{\rm{\;for every\;y}}{\rm{.}}\)

The following is true for \({\rm{x^I \{ 0,1,2\} }}\). according to the definition.

\({{\rm{p}}_{\rm{X}}}{\rm{(x) = }}\mathop {\rm{{aa}}}\limits_{{\rm{y\^I \{ 0,1,2\} }}} {\rm{np(x,y) = p(x,0) + p(x,1) + p(x,2)}}{\rm{.}}\)

The marginal probability is the sum of each row for every \({\rm{x^I \{ 0,1,2\} }}\). As a result, we've

\(\begin{array}{*{20}{c}}{{{\rm{p}}_{\rm{X}}}{\rm{(0) = 0}}{\rm{.1 + 0}}{\rm{.04 + 0}}{\rm{.02 = 0}}{\rm{.16,}}}\\{{{\rm{p}}_{\rm{X}}}{\rm{(1) = 0}}{\rm{.08 + 0}}{\rm{.2 + 0}}{\rm{.06 = 0}}{\rm{.34,}}}\\{{{\rm{p}}_{\rm{X}}}{\rm{(2) = 0}}{\rm{.06 + 0}}{\rm{.14 + 0}}{\rm{.3 = 0}}{\rm{.5,}}}\end{array}\)

which calculates \({\rm{X }}\)marginal \({\rm{pmf}}\)

Similarly, for \({\rm{Y}}\)marginal \({\rm{pmf}}\), we get

\(\begin{array}{*{20}{c}}{{{\rm{p}}_{\rm{Y}}}{\rm{(0) = 0}}{\rm{.1 + 0}}{\rm{.08 + 0}}{\rm{.06 = 0}}{\rm{.24}}}\\{{{\rm{p}}_{\rm{Y}}}{\rm{(1) = 0}}{\rm{.04 + 0}}{\rm{.2 + 0}}{\rm{.14 = 0}}{\rm{.38}}}\end{array}\)

\({{\rm{p}}_{\rm{Y}}}{\rm{(2) = 0}}{\rm{.02 + 0}}{\rm{.06 + 0}}{\rm{.3 = 0}}{\rm{.38}}\)

where we add up the probabilities in each column

To summaries, we have \({\rm{X and Y}}\)marginal \({\rm{pmf}}\), respectively, supplied with

\(\begin{aligned}{}{{{\rm{p}}_{\rm{X}}}{\rm{(0) = 0}}{\rm{.16}}}\\{{{\rm{p}}_{\rm{X}}}{\rm{(1) = 0}}{\rm{.34}}}\\{{{\rm{p}}_{\rm{X}}}{\rm{(2) = 0}}{\rm{.5,}}}\\{{{\rm{p}}_{\rm{X}}}{\rm{(x) = 0,}}\;\;\;{\rm{x"I \{ 0,1,2\} }}{\rm{.}}}\\{{{\rm{p}}_{\rm{Y}}}{\rm{(1) = 0}}{\rm{.38,}}}\\{{{\rm{p}}_{\rm{Y}}}{\rm{(2) = 0}}{\rm{.38,}}}\\{{{\rm{p}}_{\rm{Y}}}{\rm{(y) = 0,}}\;\;\;{\rm{y"I \{ 0,1,2\} }}{\rm{.}}}\end{aligned}\)

Using \({\rm{X }}\)minimal \({\rm{pmf}}\), we get

\(P(X拢1) = {p_X}(0) + {p_X}(1) = 0.16 + 0.34 = 0.5.\)

06

Determining \({\rm{X}}\) and\({\rm{Y}}\)independent \({\rm{rv's}}\)

(e) If and only if, two random variables \({\rm{X and Y}}\)are independent.

1.\({\rm{p(x,y) = }}{{\rm{p}}_{\rm{X}}}{\rm{(x) \times }}{{\rm{p}}_{\rm{Y}}}{\rm{(y)}}\)

where \({\rm{X and Y}}\)are discrete \({\rm{rv's}}\), for every \({\rm{(x,y)}}\)

2. \({\rm{f(x,y) = }}{{\rm{f}}_{\rm{X}}}{\rm{(x) \times }}{{\rm{f}}_{\rm{Y}}}{\rm{(y)}}\)

where \({\rm{X and Y}}\)are discrete \({\rm{rv's}}\), for every \({\rm{(x,y)}}\)

Otherwise, they are reliant on the following factors:

\(\begin{array}{*{20}{c}}{{{\rm{p}}_{\rm{X}}}{\rm{(2) \times }}{{\rm{p}}_{\rm{Y}}}{\rm{(2) = 0}}{\rm{.5 \times 0}}{\rm{.38 = 0}}{\rm{.19}}}\\{{\rm{p(2,2) = 0}}{\rm{.3}}}\end{array}\)

It leads us to the conclusion that the following holds true for the pair \({\rm{(2,2)}}\)

\(\begin{array}{*{20}{c}}{p(2,2) = 0.30.19 = {p_X}(2) \times {p_Y}(2)} \\{p(2,2){p_X}(2) \times {p_Y}(2)}\end{array}\)

The random variables are so dependent.

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Most popular questions from this chapter

You have two lightbulbs for a particular lamp. Let\({\rm{X = }}\)the lifetime of the first bulb and\({\rm{Y = }}\)the lifetime of the second bulb (both in\({\rm{1000}}\)s of hours). Suppose that\({\rm{X}}\)and\({\rm{Y}}\)are independent and that each has an exponential distribution with parameter\({\rm{\lambda = 1}}\).

a. What is the joint pdf of\({\rm{X}}\)and\({\rm{Y}}\)?

b. What is the probability that each bulb lasts at most\({\rm{1000}}\)hours (i.e.,\({\rm{X拢1}}\)and\({\rm{Y拢1)}}\)?

c. What is the probability that the total lifetime of the two bulbs is at most\({\rm{2}}\)? (Hint: Draw a picture of the region before integrating.)

d. What is the probability that the total lifetime is between\({\rm{1}}\)and\({\rm{2}}\)?

A shipping company handles containers in three different sizes:\(\left( 1 \right)\;27f{t^3}\;\left( {3 \times 3 \times 3} \right)\)\(\left( 2 \right) 125 f{t^3}, and \left( 3 \right)\;512 f{t^3}\). Let \({X_i}\left( {i = \;1, 2, 3} \right)\)denote the number of type i containers shipped during a given week. With \({\mu _i} = E\left( {{X_i}} \right)\)and\(\sigma _i^2 = V\left( {{X_i}} \right)\), suppose that the mean values and standard deviations are as follows:

\(\begin{array}{l}{\mu _1} = 200 {\mu _2} = 250 {\mu _3} = 100 \\{\sigma _1} = 10 {\sigma _2} = \,12 {\sigma _3} = 8\end{array}\)

a. Assuming that \({X_1}, {X_2}, {X_3}\)are independent, calculate the expected value and variance of the total volume shipped. (Hint:\(Volume = 27{X_1} + 125{X_2} + 512{X_3}\).)

b. Would your calculations necessarily be correct if \({X_i} 's\)were not independent?Explain.

Compute the correlation coefficient \({\rm{\rho }}\) for \({\rm{X}}\) and \({\rm{Y}}\)(the covariance has already been computed).

The National Health Statistics Reports dated Oct. \({\rm{22, 2008}}\), stated that for a sample size of \({\rm{277 18 - }}\)year-old American males, the sample mean waist circumference was \({\rm{86}}{\rm{.3cm}}\). A somewhat complicated method was used to estimate various population percentiles, resulting in the following values:

a. Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution.

b. Suppose that the population mean waist size is \({\rm{85cm}}\)and that the population standard deviation is \({\rm{15cm}}\). How likely is it that a random sample of \({\rm{277}}\) individuals will result in a sample mean waist size of at least \({\rm{86}}{\rm{.3cm}}\)?

c. Referring back to (b), suppose now that the population mean waist size in \({\rm{82cm}}\).Now what is the (approximate) probability that the sample mean will be at least \({\rm{86}}{\rm{.3cm}}\)? In light of this calculation, do you think that \({\rm{82cm}}\)is a reasonable value for \({\rm{\mu }}\)?

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.
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