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Chapter 3: Q121E (page 140)

Consider a collection \({{\rm{A}}_{\rm{1}}}{\rm{,}}...{\rm{,}}{{\rm{A}}_k}\) of mutually exclusive and exhaustive events, and a random variable \({\rm{X}}\) whose distribution depends on which of the \({{\rm{A}}_{\rm{i}}}{\rm{'s}}\) occurs (e.g., a commuter might select one of three possible routes from home to work, with \({\rm{X}}\) representing the commute time). Let \({\rm{E(X|}}{{\rm{A}}_{\rm{i}}}{\rm{)}}\) denote the expected value of \({\rm{X}}\) given that the event \({{\rm{A}}_{\rm{i}}}\) occurs. Then it can be shown that \({\rm{E(X) = }}\sum {{\rm{E(X|}}{{\rm{A}}_{\rm{i}}}{\rm{)}} \cdot {\rm{P(}}{{\rm{A}}_{\rm{i}}}{\rm{)}}} \), the weighted average of the individual 鈥渃onditional expectations鈥� where the weights are the probabilities of the partitioning events.

a. The expected duration of a voice call to a particular telephone number is \({\rm{3}}\) minutes, whereas the expected duration of a data call to that same number is \({\rm{1}}\) minute. If \({\rm{75\% }}\) of all calls are voice calls, what is the expected duration of the next call?

b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type \({\rm{i}}\) cookie has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{i}}}{\rm{ = i + 1 (i = 1,2,3)}}\). If \({\rm{20\% }}\) of all customers purchasing a chocolate chip cookie select the first type, \({\rm{50\% }}\) choose the second type, and the remaining \({\rm{30\% }}\) opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?

Short Answer

Expert verified

(a) The expected duration of the next call is\({\rm{2}}{\rm{.50}}\).

(b) The expected number of chips in a cookie purchased by the next customer is \({\rm{3}}{\rm{.1}}\).

Step by step solution

01

Concept Introduction

The complement rule is a statistical theorem that establishes a link between the probability of an occurrence and the probability of its complement, such that if one of these probabilities is known, automatically the other is also known.

The formula used is 鈥�

\({\rm{E(X) = }}\sum {{\rm{E(X|}}{{\rm{A}}_{\rm{i}}}{\rm{)P(}}{{\rm{A}}_{\rm{i}}}{\rm{)}}} \)

02

Duration of the next call

(a)

It is given that 鈥�

\(\begin{array}{c}{\rm{E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{1}}}} \right){\rm{ = 3}}\\{\rm{E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{2}}}} \right){\rm{ = 1}}\\{\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = 75\% = 0}}{\rm{.75}}\end{array}\)

The complement rule is represented as 鈥�

\({\rm{P(not A) = 1 - P(A)}}\)

Using the complement rule 鈥�

\(\begin{array}{c}{\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = 1 - P}}\left( {{{\rm{A}}_{\rm{1}}}} \right)\\{\rm{ = 1 - 0}}{\rm{.75 = 0}}{\rm{.25}}\end{array}\)

Using the formula 鈥�

\(\begin{array}{c}{\rm{E(X) = E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{1}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ + E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{2}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right)\\{\rm{ = 3 \times 0}}{\rm{.75 + 1 \times 0}}{\rm{.25}}\\{\rm{ = 2}}{\rm{.50}}\end{array}\)

Therefore, the value is obtained as \({\rm{2}}{\rm{.50}}\).

03

Number of chips in a cookie

(b)

It is given that 鈥�

\(\begin{array}{c}{{\rm{\mu }}_{\rm{i}}}{\rm{ = i + 1}}\\{\rm{i = 1,2,3}}\\{\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = 20\% = 0}}{\rm{.20}}\\{\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = 50\% = 0}}{\rm{.50}}\\{\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = 30\% = 0}}{\rm{.30}}\end{array}\)

Using\({{\rm{\mu }}_{\rm{i}}}{\rm{ = i + 1}}\)to determine the expected values 鈥�

\(\begin{array}{c}{\rm{E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{1}}}} \right){\rm{ = }}{{\rm{\mu }}_{\rm{1}}}{\rm{ = 1 + 1 = 2}}\\{\rm{E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{2}}}} \right){\rm{ = }}{{\rm{\mu }}_{\rm{2}}}{\rm{ = 2 + 1 = 3}}\\{\rm{E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{3}}}} \right){\rm{ = }}{{\rm{\mu }}_{\rm{3}}}{\rm{ = 3 + 1 = 4}}\end{array}\)

Using the formula 鈥�

\(\begin{array}{c}{\rm{E(X) = E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{1}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ + E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{2}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ + E}}\left( {{\rm{X}}\mid {{\rm{A}}_{\rm{3}}}} \right){\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right)\\{\rm{ = 2 \times 0}}{\rm{.20 + 3 \times 0}}{\rm{.50 + 4 \times 0}}{\rm{.30}}\\{\rm{ = 3}}{\rm{.1}}\end{array}\)

Therefore, the value is obtained as \({\rm{3}}{\rm{.1}}\).

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Most popular questions from this chapter

Show that E(X) = np when X is a binomial random variable. (Hint: First express E(X) as a sum with lower limit x =\({\rm{1}}\). Then factor out np, let y = x\({\rm{ - 1}}\)so that the sum is from y = 0 to y = n\({\rm{ - 1}}\), and show that the sum equals\({\rm{1}}\).)

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter\({\rm{\alpha }}\), the expected number of trees per acre, equal to\({\rm{80}}\). a. What is the probability that in a certain quarter-acre plot, there will be at most\({\rm{16}}\)trees? b. If the forest covers\({\rm{85,000}}\)acres, what is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius\({\rm{.1}}\)mile. Let X = the number of trees within that circular region. What is the pmf of X? (Hint:\({\rm{1}}\)sq mile\({\rm{ = 640}}\)acres.)

In proof testing of circuit boards, the probability that any particular diode will fail is\(.{\bf{01}}\). Suppose a circuit board contains\({\bf{200}}\)diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that is expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (Aboard works properly only if all its diodes work.)

Consider a disease whose presence can be identified by carrying out a blood test. Let \({\rm{p}}\) denote the probability that a randomly selected individual has the disease. Suppose \({\rm{n}}\) individuals are independently selected for testing. One way to proceed is to carry out a separate test on each of the \({\rm{n}}\) blood samples. A potentially more economical approach, group testing, was introduced during World War \({\rm{II}}\) to identify syphilitic men among army inductees. First, take a part of each blood sample, combine these specimens, and carry out a single test. If no one has the disease, the result will be negative, and only the one test is required. If at least one individual is diseased, the test on the combined sample will yield a positive result, in which case the n individual tests are then carried out. If \({\rm{p = }}{\rm{.1}}\) and \({\rm{n = 3}}\), what is the expected number of tests using this procedure? What is the expected number when \({\rm{n = 5}}\)? (The article 鈥淩andom Multiple-Access Communication and Group Testing鈥� (IEEE Trans. on Commun., \({\rm{1984: 769 - 774}}\)) applied these ideas to a communication system in which the dichotomy was active/ idle user rather than diseased/non-diseased.)

The negative binomial rv \({\bf{X}}\) was defined as the number of \({\bf{F}}\)鈥檚 preceding the rth \({\bf{S}}\). Let \({\bf{Y}}{\rm{ }}{\bf{5}}\) the number of trials necessary to obtain the rth \({\bf{S}}\). In the same manner in which the pmf of \({\bf{X}}\) was derived, derive the pmf of \({\bf{Y}}\)
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