Statistik Chapter 4

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Statistik Chapter 4

  1. 1. Chapter Outline<br />4.1Random Variable<br />Discrete Random Variable<br />Probability Distribution of a Discrete Random Variable<br />4.3.1Mean of a Discrete Random Variable<br />4.3.2Standard Deviation of a Discrete Random Variable<br />4.4Cumulative Distribution Function for Discrete Random Variable<br />4.5Continuous Random Variable<br />Objectives<br />After completing this chapter, you should be able to:<br />Interpret that a random variable is a numerical quantity whose value depends on the conditions and probabilities associated with an experiment.<br />Differentiate between a discrete and a continuous random variable.<br />Construct a discrete probability distribution based on an experiment or given function.<br />Determine the similarities and differences between frequency distributions and probability distribution.<br />Compute, describe and interpret the mean and standard deviation of a probability distribution.<br />Random Variable<br />Definition:<br />A random variable is a variable whose value is determined by the outcome of a random experiment.<br />Supposed one family is randomly selected from the population. The process of random selection is called random or chance experiment.<br />Let X be the number of vehicles owned by the selected family (0, 1, 2, …, n). Therefore the first column represents five possible values (0, 1, 2, 3 and 4) of vehicles owned by the selected family.<br />This table shows that 30 families have 0 vehicle, 470 families have 1 vehicle, 850 families have 2 vehicles, 490 families have 3 vehicles and 160 families have 4 vehicles.<br />In general, a random variable is denoted by X or Y.<br />Discrete Random Variable<br />Definition:<br />A random variable that assumes countable values is called discrete random variable.<br />Examples of discrete random variables:<br />Number of houses sold by a developer in a given month.<br />Number of cars rented at a rental shop during a given month.<br />Number of report received at the police station on a given day.<br />Number of fish caught on a fishing trip.<br />Probability Distribution of a Discrete Random <br /> Variable<br />Definition:<br />The probability distribution of a discrete random variable lists all the possible values that the random variable can assume and their corresponding probabilities.<br />It is used to represent populations.<br />The probability distribution can be presented in the form of a mathematical formula, a table or a graph. <br />Example 1<br />Consider the table below. Let X be the number of vehicles owned by a randomly selected family. Write the probability distribution of X and graph for the data.<br />Solution:<br />Example 2<br />During the summer months, a rental agency keeps track of the number of chain saws it rents each day during a period of 90 days and X denotes the number of saws rented per day. Construct a probability distribution and graph for the data.<br />XNumber of days045130215Total90<br />Solution:<br />When, <br />Hence, the probability distribution for X:<br />Whereas the graph is shown below:<br />Example 3<br />One small farm has 10 cows where 6 of them are male and the rest are female. A veterinary in country XY wants to study on the foot and mouth disease that attacks the cows. Therefore, she randomly selects without replacement two cows as a sample from the farm. Based on the study, construct a probability distribution which X is the random sample representing the number of male cows that being selected as a sample (use tree diagram to illustrate the above event).<br />MFMFMFJoint ProbabilityP(MM)=P(MF)=P(FM)=P(FF)=<br />XP(x)<br />Conditions for probabilities for discrete random variable.<br />The probability assigned to each value of a random variable x must be between 0 and 1.<br />0 P(x) 1,for each value of x.<br />The sum of the probabilities assigned to all possible values of x is equal to 1.<br />P(x) = 1<br />Example 4<br />The following table lists the probability distribution of car sales per day in a used car shop based on passed data.<br />Car Sales per day, X0123P(x)0.100.250.300.35<br />Find the probability that the number of car sales per day is,<br />none<br />exactly 1<br />1 to 3<br />more than 1<br />at most 2<br />4.3.1 Mean of a Discrete Random Variables<br />Definition:<br />The mean of a discrete random variable X is the value that is expected to occur repetition, on average, if an experiment is repeated a large number of times. It is denoted by and calculated as:<br /> <br />The mean of a discrete random variable X is also called as its expected value and is denoted by E(X),<br />4.3.2 Standard Deviation of a Discrete Random Variable<br />Definition:<br />The standard deviation of a discrete random variable X measures the spread of its probability distribution and is calculated as:<br /> <br />A higher value for the standard deviation of a discrete random variable indicates that X can assume value over a large range about the mean. <br />In contrast, a smaller value for the standard deviation indicates the most of the value that X can assume clustered closely about the mean.<br />Example 5<br />The following table lists the probability distribution of car sales per day in a used car dealer based on passed data. P(x) is the probability of the corresponding value of X = x. Calculate the expected number of sales per day and followed by standard deviation.<br />XP(x)00.110.2520.330.35Total1.00<br />Solution:<br />Mean<br />Standard Deviation<br />Example 6 <br />During the summer months, a rental agency keeps track of the number of chain saws it rents each day during a period of 90 days and X denotes the number of saws rented per day. What is the expected number of saws rented per day? Then, find the standard deviation.<br />X012P(x)0.50.330.17<br />Solution:<br />Mean<br /> <br />Standard Deviation<br />Cumulative Distribution Function<br />Definition:<br />The cumulative distribution function (CDF) for a random variable X is a rule or table that provides the probabilities for any real number x. <br />Generally the term cumulative probability refers to the probability that X less than or equal to a particular value.<br />For a discrete random variable, the cumulative probability is a function , <br />where <br />and <br />, where is the probability distribution function for X.<br />Example 7<br />A discrete random variable X has the following probability distribution.<br />X0123<br />Construct the cumulative distribution of X.<br />Solution:<br />X0123P(x)F(x)<br />Example 8 <br />A discrete random variable X has the following cumulative distribution.<br />a) Construct the probability distribution of X.<br />X012345P(x)F(x)<br />b) Construct the graph of the:<br />i.probability distribution of X.<br />cumulative distribution of X.<br />Example 9 (Overall Example)<br />During the school holiday, the manager of Victory Hotel records the number of room bookings being cancelled each day during a period of 50 days, the results are shown below and X denotes the number of room bookings being cancelled per day.<br />Number of room bookings being cancelled per day, XNumber of days021427384135106373<br />Construct the probability distribution of X. <br />X01234567P(x)<br />Then, draw a bar chart for the probability distribution.<br />X0.040.080.120.200.16012P(x)30.24450.2867<br />The manager expects that five room bookings were cancelled for a day. Is the manager expectation true? Explain.<br />The manager expectation is not true since only four expected room bookings being cancelled for a day.<br />Find the probability that at most three room bookings were cancelled.<br />Find the standard deviation for the number of room bookings being cancelled.<br />X01234567P(x)0.140.160.060.06X2.P(x)00.084.165<br />Continuous Random Variable<br />Definition:<br />A random variable that can assume any value contained in one or more intervals is called a continuous random variable.<br />Examples of continuous random variables,<br />The weight of a person.<br />The time taken to complete a 100 meter dash.<br />The duration of a battery.<br />The height of a building.<br />EXERCISES<br />1.The following table gives the probability distribution of a discrete random variable X.<br />X012345P(x)0.30.170.220.310.150.12<br />Find the following probability:<br />a)exactly 1.<br />b)at most 1.<br />c)at least 3.<br />d)2 to 5.<br />e)more than 3.<br />2.The following table lists the frequency distribution of the data collected by a local research agency.<br />Number of TV sets own0123456Number of families11089132934015176103<br />a)Construct the probability distribution table.<br />b)Let X denote the number of television sets owned <br />by a randomly selected family from this town. Find the following probabilities:<br />i.exactly 3.<br />ii.more than 2.<br />iii.at most 2.<br />iv.1 to 3.<br />v.at least 4.<br />3.According to a survey 65% university students smokes. Three students are randomly selected from this university. Let X denote the number of students in this sample who does not smokes. Develop the probability distribution of X.<br />a)Draw a tree diagram for this problem.<br />b)Construct the probability distribution table.<br />c)Let X denote the number of students who does. <br />not smoking is selected randomly. Find the following probability:<br />i.at most 1.<br />ii.1 to 2.<br />at least 2.<br />more than 1.<br />4.The following table gives the probability distribution of the number of camcorders sold on a given day at an electronic store.<br />Camcorder sold0123456Probability0.050.120.190.300.180.100.06<br />Calculate the mean and standard deviation for this probability distribution.<br />5.According to a survey, 30% of adults are against using animals for research. Assume that this result holds true for the current population of all adults. Let x be the number of adults who agrees using animals for research in a random sample of three adults. Obtain: <br />a)the probability distribution of X.<br />b)mean.<br />standard deviation.<br />In a genetics investigation, cat litters with ten kittens are studied which of three are male. The scientist selects three kittens randomly. Let X as the number of female kittens that being selected and construct probability distribution for X (you may use tree diagram to represent the above event). Based on the probability distribution obtained, find the:<br />mean.<br />standard deviation.<br />An urn holds 5 whites and 3 black marbles. If two marbles are drawn randomly without replacement and X denoted the number of white marbles,<br />Find the probability distribution of X.<br />Plot the cumulative frequency distribution (CFD) of X.<br />The following table is the probability distribution for the number of traffic accidents occur daily in a small city. <br />Number of accidents (X)012345P(x)0.100.209a3aaa<br />Find the probability of:<br />exactly three accidents occur daily.<br />between one and four accidents occur daily.<br />at least three accidents occur daily.<br />more than five accidents occur daily and explain your answer.<br />Traffic Department of that small city expects that 5 accidents occur daily. Do you agree? Justify your opinion.<br />Compute the standard deviation.<br />The manager of large computer network has developed the following probability distribution of the number of interruptions per day:<br />Interruptions(X)0123456P(x)0.320.350.180.080.040.020.01<br />Find the probability of:<br />more than three interruptions per day.<br />from one to five interruptions per day.<br />at least an interruption per day.<br />Compute the expected value.<br />Compute the standard deviation.<br />You are trying to develop a strategy for investing in two different stocks. The anticipated annual return for a RM1,000 investment in each stock has the following probability distribution.<br />Returns (RM), XP(x)Stock AStock B-100500.101500.380-200.3150-100a<br />Find the value of a.<br />Compute,<br />expected return for Stock A and Stock B.<br />standard deviation for both stocks.<br />Would you invest in Stock A or Stock B? Explain.<br />11.Classify each of the following random variables as discrete or continuous.<br />The time left on a parking meter.<br />The number of goals scored by a football player.<br />The total pounds of fish caught on a fishing trip.<br />The number of cans in a vending machine.<br />The time spent by a doctor examining a patient.<br />The amount of petrol filled in the car.<br />The price of a concert ticket.<br />

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