TUTORIAL- RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1

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TUTORIAL- RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. A

manufacturer of laptop computer monitors has determined that on an average, 3% of screens produced are defective. A sample of one dozen monitors from a production lot was taken at random. What is the probability that in this sample fewer than defectives will be found? 2. On any given day, the demand for refrigerators by an appliance store can be regarded as a function of a random variable. The retailer has determined that on some days he makes no sale at all and the maximum number of refrigerators sold on any given day is 7. Considering X variable as the demand, the probability distribution of demand is provided in the following table: Demand (X) Probability 0 0.05 1 0.10 2 0.15 3 0.25 4 0.20 5 0.10 6 0.10 7 0.05 1.00 Calculate the expected demand, E(X), per day. Calculate also the variance for this probability distribution. 3. A student is given 4 True or False questions. The student does not know the answer to any of the questions. He tosses a fair coin. Each time he gets a head, he selects True. What is the probability that he will get: a) Only one correct answer. b) At most 2 correct answers. c) At least 3 correct answers. d) All correct answers. 1. Newly married couple plans to have 5 children. An astrologist tells them that based on astrological reading, they have an 80% chance of having a baby boy on any particular birth. The couple would like to have 3 boys and 2 girls. Find the probability of this event. 2. The number of cars pulling into a petrol pump follows a Poison distribution with a mean of three cars every ten minutes. What is the probability that exactly two cars will arrive in the next ten minutes? 3. An automatic machine makes paper clips from coils in wire. On an average, one in 400 paper clips is defective. If the paper clips are packed in small boxes of 100 clips each, what is the probability that any given box of clips contains: a) No defectives b) One or more defectives. c) Less than two defectives. d) Two or less defectives. 4. Because of recycling campaigns, a number of empty glass soda bottles are being returned for refilling. It has been found that 10 per cent of the incoming bottles are chipped and hence are discarded. In the next batch of 20 bottles, what is the probability that: a) None will be chipped. b) Two or fewer will be chipped. c) Three or more will be chipped. d) What is the expected number of chipped bottles in a batch of 20? 5. The local telephone switchboard in a small village receives on an average five calls per hour for information. Assuming that these calls follow Poison distribution, what is the probability that: a) More than half an hour elapses between two successive calls. b) In a particular hour, five calls are received. 6. A certain hospital annually admits 50 patients per day. On an average, 3 per cent of incoming patients require rooms provided with special facilities. On the morning of a given day, it is found that three rooms with special facilities are available. Assuming that 50 patients will be admitted on that day, what is the probability that more than three patients will require special rooms? 7. An automatic machine produces 100 spools of brass wire per hour. Studies have shown that on an average, three spools of wire turn out to be defective among these 100 spools. Assuming a Poison distribution, find the probability of the machine producing the following number of defective spools per hour. a) Exactly three. b) Three or fewer. c) More than three. d) Less than two or more than three. e) What is the probability that the machine will produce seven defective spools in a period of two hours? 8. The IQ scores of students are normally distributed with a mean µ and standard deviation σ. Find the areas under the curve over the following intervals from the table of z scores. a) Area to the right of Z=2.58 b) Area to the left of Z=-(2.33) c) Area between Z=1 and Z=1.96 d) Area between Z=0 and Z=1.2 e) Area between Z=-1.28 and Z=1.28 f) Area between Z=-1.96 and Z= 1.96 9. The price of one gallon bottle of milk is normally distributed in Loliondo district with an average price of $2.00 and a standard deviation of $ 0.2. A family driving within the district stops at a Mzee wa Kikombe to buy a gallon of milk. What is the probability that they will pay: a) More than $2.10 b) Less than $ 1.75 c) Between $ 1.85 and $ 2.15 d) Between $ 1.85 and $ 1.95 10. The heights of soldiers is normally distributed with a mean of 68 inches and a variance of 9 square inches. What is the probability that a soldier picked up at random is: a) Less than 5 feet 1 inch tall. b) Between 63 inches and 66 inches. c) Taller than 6 feet. d) What must the height of the soldier be so that only 30% of the soldiers are taller than him? e) What should the height of the doorway be so that 70% of the soldiers have to duck when entering the room? 11. The average time a subscriber spending in browsing in the UCC internet cafe is 49 minutes. Assume the standard deviation is 16 minutes and the time is normally distributed. a) What is the probability a subscriber will spend at least 1 hour browsing in the cafe? b) What is the probability a subscriber will spend no more than 30 minutes browsing in the cafe? c) For the 10% who spend the most time browsing in the cafe, how much time do they spend? 12. The data of micro enterprises in Tanzania are not well established. Due to scant data, the government of Tanzania has got difficulties in establishing average tax that should be charged in this sector. Despite this limitation, one of the recent study found that in any given month the owners of micro enterprises get less than Tshs 15,000 as a profit before tax on 30% of the days and more than Tshs 40,000 on 10% of the days per month. If it known that the profit before tax is normally distributed, find the mean and variance of the profit before tax per month in this sector. What would be average tax if the tax is charged at 30% of the profit before tax? 13. The Introduction to Business class has 200 students. Studies have shown that on an average 15 per cent of students have always been absent. What is the probability that on a given day: a) 150 or more students will attend the class. b) 190 or fewer students will attend. c) Between 170 and 185 students will attend. 14. A large department store has 4500 accounts receivables. The dollar amounts of these accounts are known to be normally distributed with a mean of $150 and a standard deviation of $25. a) How many accounts are expected to be between $115 and $165 b) How many accounts are either less than $100 or more than $200. c) How many accounts are between $190 and $125 d) How many accounts are between $190 and $200 e) What is the dollar amount so that 10 per cent of accounts receivables exceed this amount? "When it is not in our power to determine what is true, we ought to follow what is most probable" Descartes