Test 1 with Answers - Probability and Statistics in Electrical Engineering | STAT 4714, Exams of Probability and Statistics

Download Test 1 with Answers - Probability and Statistics in Electrical Engineering | STAT 4714 and more Exams Probability and Statistics in PDF only on Docsity! STAT 4714, Test I Samples, Fall 2008 STAT 4714 Test I February 14, 2002 Name____________________________ 1. If you buy a ticket in a particular state lottery you must select 6 distinct numbers from the 50 numbers 1, 2,…, 50. After the sales of tickets close, prizes are awarded by drawing 6 distinct winning numbers from the set 1, 2,…, 50. (a) You win the grand prize if the 6 numbers that you selected are the same as the six winning numbers drawn. The order of the numbers is not considered. If you buy one lottery ticket, find the probability that you win the grand prize. Note that there may be more than one person who wins the grand prize, and this affects the amount of money you would receive from winning, but not your probability of winning. (b) You win a smaller prize if exactly 5 of your numbers correspond to the winning numbers drawn. If you buy one lottery ticket, find the probability that you win the smaller prize (c) Given that you did not win the grand prize, find the conditional probability that you win the smaller prize. 2. The number of system failures of a certain type that occur at a computer facility in a month has a Poisson distribution with parameter λ = 0.5. (a) Find the probability that more than two system failures occur in a month. (b) Find the probability that in a year (12 months) there is at least one month with more than two system failures. (c) Find the expected number of months in a year with more than two system failures. (d) Find the probability that the total number of system failures in a year is at least 5. 3. The proportion of time X that a computer system is in use in a 24-hour day is a continuous random variable with density function f(x) = 12x2(1 - x) for 0 ≤ x ≤ 1, and f(x) = 0 otherwise. (a) Find the cumulative distribution function F(x) for X. (b) Find the probability that the proportion of time in use is above .90. (c) Find the mean and variance of X. (d) If successive days are independent, find the probability that the tenth day is the first to have a proportion of time in use above .90. 4. Suppose that components produced by a process are independent and each has probability 0.1 of being defective. (a) Let X represent the number of defective components when 2 items are selected at random from the output of the process. Find )( xXP = for x = 0, 1, and 2. (b) When components are inspected there is the possibility that the person doing the inspection will make a mistake and misclassify the component. Suppose that when a component is defective it will be correctly classified as defective with probability 0.95 and misclassified as nondefective with probability 0.05. Also suppose that when a component is nondefective it will be correctly classified as nondefective with probability 0.98 and misclassified as defective with probability 0.02. Assume that an error in classifying one component is independent of errors in classifying other components. Let Y represent the number of components (in the 2 components in part (a)) that are classified correctly. Find )2( =YP . Note that Y = 2 does not imply that X must be any particular value. STAT 4714 Test I February 16, 2001 Name____________________________ 1. Eight applicants for three job openings are ranked according to ability, with number 1 being the best, number 2 the second best, and so on. Suppose that these rankings are unknown to the personnel manager of the employer, who simply hires three applicants at random. a) Find that probability that the best applicant is not selected in the three applicants that are hired. b) Given that the best applicant is not selected, find the conditional probability that the second best applicant is selected. 2. The probability that an experimental Local Area Network sends a corrupted data packet is .0004. a) For the next 10,000 independent data packets sent by the network, find the expected number of data packets that are corrupted. b) For the next 10,000 independent data packets sent by the network, find the approximate probability that at least 4 packets are corrupted. c) Suppose that data packets will be examined until a corrupted packet is found. Find the expected number of packets that must be examined to find the first corrupted packet. 3. Suppose that the time required to transmit an e-mail message is normally distributed with a mean of 0.72 seconds and a standard deviation of 0.10 seconds. a) Find the probability that more than 0.90 seconds are required to transmit a message. b) Find a value t such that the probability of requiring more than t seconds is only 0.01. c) Find the probability that at least one of the next ten independent messages requires more than t seconds for transmission. 4. Let the random variable X represent the life length (in months) of an electronic component, and suppose that X has an exponential distribution with density 0 ,)( >= − xexf xX αα , and moment generating function α α α < − = t t tM X ,)( . a) Use the moment generating function of X to find the mean and variance of X. b) If 01.=α find the failure rate function )(tρ . c) The manufacturer of these components would like to find a value 0x such that 90% of the components last at least 0x months. If 01.=α find 0x . STAT 4714 Test I September 25, 1998 Name_____________________________ 1. In many computer systems, memory is divided into units called pages. Pages at any given time are marked either “empty” or “in use”, with a list of these conditions maintained by the computer’s operating system (OS) indicating the status of each page. When a new user program is loaded into memory, the OS needs to find enough empty pages to accommodate it. Suppose that the probability of a page being empty is p and the pages are independent. Consider the situation in which a user program requires only one page, and the OS checks its page list sequentially until it finds an empty page. a) If p = 0.1, find the probability that exactly 5 pages must be checked. b) If p = 0.1, find the probability that more than 5 pages must be checked. c) If p = 0.1, find the expected number of pages that must be checked. 2. The length X of an injected-molded case that holds magnetic tape is should to be 90 mm, but variations in the injection molding process produce variations in the lengths of the cases. Suppose that the distribution of length is normal with mean μ = 90 mm and standard deviation σ = 0.1 mm. a) Find two values x1 and x2 such that 98% of the case lengths will be between x1 and x2. b) Find the probability that each of the next 10 cases produced will have a length between x1 and x2. c) Suppose that the specifications for the case require that the length of a case be between 89.9 mm and 90.1 mm. With the current injection molding process many of the cases will not meet this specification. To have a high proportion of the cases meet this specification, it would be necessary to reduce the variability of the injection molding process. This means that σ must be reduced below 0.1. What value of σ would be required to give a probability of .98 of meeting the specification? 3. A system has 4 components of the same type. The system will function if and only if all 4 of these components are functioning. Suppose that in constructing the system 4 components are selected at random from 50 components that are available. a) If 5 of the 50 available components are defective and will not function, find the probability that the system will not function. b) If the system will not function then at least one of the 4 components in the system must be nonfunctioning. Given that the system will not function, find the conditional probability that exactly two of the components are nonfunctioning. As in part (a), assume that there were 5 nonfunctioning components among the 50 components from which the 4 system components were selected. 4. With large databases it is usually desirable to reduce storage requirements, and one way to do this is through Huffman coding. Consider the following simple illustration of Huffman coding. Suppose that a certain database consists only of various combinations of the letters A, B, C, and D, and these letters appear with probabilities .70, .20, .05, and 0.05, respectively. If we used the two-bit code 00, 01, 10, and 11 for these four letters, then each letter would require two bits of storage space. However, an alternate storage scheme based on Huffman coding would be to have a single bit 0 represent A, the pair of bits 10 represent B, and the triples 110 and 111 represent C and D, respectively. Then the number of bits X required to store a letter would be a random variable with possible values 1, 2, and 3, and the corresponding probabilities would be .70, .20, and .10, respectively. a) Find E(X), the expected number of bits required to store a letter using this Huffman coding scheme. Is E(X) less than the 2 bits required when the code 00, 01, 10, and 11 is used? If E(X) is below 2 then this would imply that this Huffman coding scheme would be expected to reduce storage requirements. b) Find the variance of X. c) Find the moment generating function of X. d) If two letters are to be stored using the Huffman coding scheme, find the probability that exactly 4 bits are required. Assume that the two letters are independent. STAT 4714 Test I February 13, 1998 Name_____________________________ 1. In a computer system, user names consist of two letters followed by two digits. The letters are selected from the standard 26 letter alphabet of A through Z, and the digits from the 10 digits 0 through 9. The user name may repeat letters or digits. For example, GG44 is possible. a) If a user name is selected at random from all possible user names, find the probability that the two letters in this user name are the same. b) Given that the two letters in this user name are the same, find the conditional probability that the two digits are also the same. c) Given that the two letters in this user name are the same, find the conditional probability that the two digits are 44. 2. In a data communication system, several messages that arrive at a node are bundled together into a packet before they are transmitted over the network. Assume that messages arrive at the node according to a Poisson process with a mean rate of λ = 30 messages per minute. Six consecutive messages are used to form a packet. a) If this system is observed from some starting time labeled zero, find the expected time until the first message arrives at the node. b) Find the probability that a packet is not formed in the first 10 seconds. c) Find the probability that two or more packets are formed in the next 10 seconds. d) Find the probability that there are exactly 6 messages in each of the next three consecutive 10 second intervals. 3. When sending a message over a communications channel there is the possibility that the message received will not correspond exactly to the message sent because of errors produced by noise in the system. Suppose that a message consisting of a sequence of the digits 0 and 1 is to be sent over the channel. Assume that when a digit is sent, there is a probability p that there is an error in receiving this digit, and assume that errors in different digits are independent. Unless p is very small, it may be desirable to use a strategy to improve on the probability of receiving the message correctly. One approach to improving this probability is to send each digit three times and use majority decoding. For example, if we want to send 1001 then we would send 111 000 000 111. Majority decoding means that a when a digit is sent three times, then the digit is decoded as 1 if 1 is in the majority, and as 0 if 0 is in the majority. For example, if we send 111 for the first digit in the message and receive 101, then this will be decoded as 1, but if 100 is received then this is decoded as 0. In the second case the decoded digit is incorrect. Note that when a digit is sent three times, each of the three individual transmissions has probability p of an error. a) If a digit is sent three times and p = .01, find the probability that the decoded digit will be correct. Note that this probability does not depend on whether 0 or 1 is sent. b) Suppose that a message consisting of n = 100 digits is to be sent, and each of the 100 digits will be sent three times with majority decoding used. Find the probability that at least one of the 100 decoded digits will be incorrect. 4. Suppose that the continuous random variable X represents the current in a thin copper wire in milliamperes, and that the density function of X is . f x x x x( ) . . ,= − ≤ ≤06 006 0 102 a) Find E(X), the expected current in the wire. b) Find P(X < 2), the probability that the current will be less than 2 milliamperes. c) Suppose that the current is measured at five different times, and the current readings at these times are independent. Find the probability that the current will be less than 2 milliamperes at exactly two of these five times. 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