Probability and Statistics

Problem 6: The velocity of a particle, v km/s, after t seconds is given by: 25 v = 20 – + for 1 st 5 10 a) Find an expression for the acceleration of the particle in km/s, in terms of t. b) Calculate the particle’s maximum speed in km/s, show all parts of your solution. Problem 7: The Venn diagram shows customers’ choice of cheese (C), tuna (T) and egg (E) fillings for a sandwich in a café. с 6 5 12 15 23 1 E a) How many people chose all three fillings? b) If a customer is chosen at random; calculate the following probabilities: i. P(TIC) ii. P(TE) iii. P(CE)

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Problem 8: A box contains two black stones and four white stones. One is randomly selected and not replaced before another is randomly selected. a) Draw a tree diagram to show all the possible outcomes. b) Calculate the probability of selecting: i. A black stone and a white stone, in that order. ii. A black stone and a white stone, in any order. iii. At least one black stone. Problem 9: ABCDEF is a regular hexagon. AB = a, BC = b and FC 2a А а в b 2a F с E D х a) Find in terms of a and b, a representation of the following vectors: i. FE ii. CE b) Knowing that CE = EX prove that that FX is parallel to CD

EXPERT ANSWER There are 2 black stones and 4 white stones. a)The tree diagram ahowing possible outcomes is given below. b) i)The number of permutations of selecting 2 from 2+4=6 (order important) is  . The number of possible ways of selecting a black stone and then a white stone (order important) is  . The required probability is  . …

Problem 8: A box contains two black stones and four white stones. One is randomly selected and not replaced before another is randomly selected. a) Draw a tree diagram to show all the possible outcomes. b) Calculate the probability of selecting: i. A black stone and a white stone, in that order. ii. A black stone and a white stone, in any order. iii. At least one black stone. Problem 9: ABCDEF is a regular hexagon. AB = a, BC = b and FC 2a А а в b 2a F с E D х a) Find in terms of a and b, a representation of the following vectors: i. FE ii. CE b) Knowing that CE = EX prove that that FX is parallel to CD Read More »

During the summer months, a rental agency keeps track of the number of chain saws it rents each day for a period of 90 days. The number of saws rented per day is represented by the variable X. The results are shown here. Compute the probability P(X) for each X. Х 0 Number of days 45 20 25 90 Total Data/local/Packages/Microsoft MicrosoftEdge_8wekyb3d8bbwe/TempState/Downloads/Mat%20119%20and%20Mat%20120(1 + 2 Fit to page Page view Two requirements for a Probability Distribution The sum of the probabilities of all the events in the sample space must equal 1. That is, P(x)=1 2. The probability of each event in the sample space must be between or equal to 0 and 1. That is, 0 5 P(x) 31. Example 1 Toss a fair, six-sided die. Let X the number of face that show. a) Construct a probability distribution for toss a fair, six-sided die. b) Is it probability distribution function(PDF)? Why? Why not? Example 2 Construct a probability distribution for a family of three children. Let Y equal the number of girls. Example 3 During the summer months, a rental agency keeps track of the number of chain saws it rents each day for a period of 90 days. The number of saws rented per day is represented by the variable X The results are shown here. Compute the probability P(X) for each X Х 0 Number of days 45 20 25 90 2 Total – + 1 Fit to page CD Page view I A 0 Determine whether each distribution is a probability distribution (PDF). a) P(x) 1/5 1/5 10 1/5 15 1/5 20 1/5 5 b) 1 2 3 4 P(v) 14 1/8 1/16 9/16 c) b 0 1 P(b) – 1.0 0.7 0.3 0.2 2 3 d) 2 2 3 7 P(x) 0.5 0.3 0.4

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29. The time to failure in hours of an electronic component subjected to an accelerated life test is shown in Table 1. To accelerate the failure test, the units were tested at elevated temperature (read down, then across). (Must show how each is calculated) (a) Calculate the sample average and standard deviation. (b) Construct a histogram (c) Find the sample median and the lower and upper quartiles. Table 1 Electronic Component Failure Time 127 125 131 124 129 121 142 151 160 125 124 123 120 119 128 133 137 124 142 123 121 136 140 137 125 124 128 129 130 122 118 131 125 133 141 125 140 131 129 126

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Exercise: Neon lights Neon lights on the U of A campus are replaced at the average rate of 100 units per day; in fact, the daily demand for new lights is normal with mean 10o and standard deviation 10 units. The physical plant orders neon lights periodically. It costs $100 to initiate a purchase order. A neon light kept in storage is estimated to cost $o.02 per day. The lead time between placing and receiving an order is 2 days. Determine the optimal ordering policy such that the probability of running out of stock is 0.05

EXPERT ANSWER D = Demand per day = 100 (the replacement per day) S = Ordering cost = $100 h = Holding cost per item per day = $0.02 L = Average lead time = 12 We will apply EOQ formula to determine the optimal order quantity Q*.  = 1000 Reorder point = R = …

Exercise: Neon lights Neon lights on the U of A campus are replaced at the average rate of 100 units per day; in fact, the daily demand for new lights is normal with mean 10o and standard deviation 10 units. The physical plant orders neon lights periodically. It costs $100 to initiate a purchase order. A neon light kept in storage is estimated to cost $o.02 per day. The lead time between placing and receiving an order is 2 days. Determine the optimal ordering policy such that the probability of running out of stock is 0.05 Read More »

29. The time to failure in hours of an electronic component subjected to an accelerated life test is shown in Table 1. To accelerate the failure test, the units were tested at elevated temperature (read down, then across). (Must show how each is calculated) (a) Calculate the sample average and standard deviation. (b) Construct a histogram (c) Find the sample median and the lower and upper quartiles. Table 1 Electronic Component Failure Time 127 125 131 124 129 121 142 151 160 125 124 123 120 119 128 133 137 124 142 123 121 136 140 137 125 124 128 129 130 122 118 131 125 133 141 125 140 131 129 126

EXPERT ANSWER b) Histogram: