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Chapter 3: Q5E (page 98)
If the sample space S is an infinite set, does this necessarily imply that any rv X defined from S will have an infinite set of possible values? If yes, say why. If no, give an example.
No. Consider the random variable X defined as 鈥楪etting a Head鈥� in a random experiment of flipping a coin n times.
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A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that
a. At most 6 of the calls involve a fax message?
b. Exactly 6 of the calls involve a fax message?
c. At least 6 of the calls involve a fax message?
d. More than 6 of the calls involve a fax message?
A new battery鈥檚 voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of batteries that must be tested.
a. What is\(p\left( 2 \right)\), that is, \(P\left( {Y = 2} \right)\),?
b. What is\(p\left( 2 \right)\)? (Hint: There are two different outcomes that result in\(Y = 3\).)
c. To have \(Y = 5\), what must be true of the fifth battery selected? List the four outcomes for which Y = 5 and then determine\(p\left( 5 \right)\).
d. Use the pattern in your answers for parts (a)鈥�(c) to obtain a general formula \(p\left( y \right)\).
A library subscribes to two different weekly news magazines, each of which is supposed to arrive in Wednesday鈥檚 mail. In actuality, each one may arrive on Wednesday, Thursday, Friday, or Saturday. Suppose the two arrive independently of one another, and for each one\(P\left( {Wed.} \right) = 0.3\), \(P\left( {Thurs.} \right) = 0.4\), \(P\left( {Fri.} \right) = 0.2\), and\(P\left( {Sat.} \right) = 0.1\). Let Y = the number of days beyond Wednesday that it takes for both magazines to arrive (so possible Y values are 0, 1, 2, or 3). Compute the pmf of Y. (Hint: There are 16 possible outcomes; \(Y\left( {W,W} \right) = {\bf{0}}\),\(Y\left( {F,Th} \right) = 2\), and so on.)
Grasshoppers are distributed at random in a large field according to a Poisson process with parameter a \({\rm{\alpha = 2}}\) per square yard. How large should the radius \({\rm{R}}\) of a circular sampling region be taken so that the probability of finding at least one in the region equals \({\rm{.99}}\)?
A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table.
X | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
p(x) | .10 | .15 | .20 | .25 | .20 | .06 | .04 |
Calculate the probability of each of the following events.
a. {at most three lines are in use}
b. {fewer than three lines are in use}
c. {at least three lines are in use}
d. {between two and five lines, inclusive, are in use}
e. {between two and four lines, inclusive, are not in use}
f. {at least four lines are not in use}
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