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RANDOM VARIABLES • Random variables • Probability distribution • Random number generation – Expected value – Variance – Probability distributions 1 RANDOM VARIABLES • Random variable: – A variable whose numerical value is determined by the outcome of a random experiment • Discrete random variable – A discrete random variable has a countable number of possible values. – Example • Number of heads in an experiment with 10 coins • If X denotes the number of heads in an experiment with 10 coins, then X can take a a value of 0, 1, 2, …, 10 2 RANDOM VARIABLES – Other examples of discrete random variable: number of defective items in a production batch of 100, number of customers arriving in a bank in every 15 minute, number of calls received in an hour, etc. • Continuous random variable – A continuous random variable can assume an uncountable number of values. – Examples • The time between two customers arriving in a bank, the time required by a teller to serve a customer, etc. 3 DISCRETE PROBABILITY DSTRIBUTION • Discrete probability distribution – A table, formula, or graph that lists all possible events and probabilities a discrete random variable can assume – An example is shown below: Discrete Probability Distribution Probability 0.75 0.5 0.25 0 HH HT Event TT 4 CONTINUOUS PROBABILITY DSTRIBUTION • Continuous probability distribution – Similar to discrete probability distribution – Since there are uncountable number of events, all the events cannot be specified – Probability that a continuous random variable will assume a particular value is zero!! – However, the probability that the continuous random variable will assume a value within a certain specified range, is not necessarily zero – A continuous probability distribution gives probability values for a range of values that the continuous random variable may assume 5 f(x) CONTINUOUS PROBABILITY DSTRIBUTION z 6 f(x) CONTINUOUS PROBABILITY DSTRIBUTION z 7 REVISIT SIMPLE RANDOM SAMPLING • In Chapter 5, a simple random sample of 10 families is chosen from a group of 40 families. – 40 Random numbers are generated – Each random number is between 0 and 1 (not including 1) – Excel RAND() function is used to generate each random number. 8 REVISIT SIMPLE RANDOM SAMPLING – What is the average of the random numbers generated? – What is the variance of the random numbers generated? – What is the standard deviation of the random numbers generated? E 9 Average 10 Variance 11 Standard deviation F 0.5155 0.0856 0.292635444 9 REVISIT SIMPLE RANDOM SAMPLING – Plot a histogram with all the random numbers, and comment on the distribution of the random numbers. Frequency Histogram 8 7 6 5 4 3 2 1 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 Random Numbers 1 10 RANDOM NUMBER GENERATION • Most software can generate discrete and continuous random numbers (these random numbers are more precisely called pseudo random numbers) with a wide variety of distributions • Inputs specified for generation of random numbers: – Distribution – Average – Variance/standard deviation – Minimum number, mode, maximum number, etc. 11 RANDOM NUMBER GENERATION • Next 4 slides – show histograms of random numbers generated and corresponding input specification. – observe that the actual distribution are similar to but not exactly the same as the distribution desired, such imperfections are expected – methods/commands used to generate random numbers will not be discussed in this course 12 RANDOM NUMBER GENERATION: EXAMPLE – A histogram of random numbers: uniform distribution, min = 500 and max = 800 Frequency Uniform Distribution 25 20 15 10 5 0 Random Numbers 13 RANDOM NUMBER GENERATION: EXAMPLE – A histogram of random numbers: triangular distribution, min = 3.2, mode = 4.2, and max = 5.2 Frequency Triangular Distribution 120 100 80 60 40 20 0 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 Radom Numbers 5 5.2 14 RANDOM NUMBER GENERATION: EXAMPLE – A histogram of random numbers: normal distribution, mean = 650 and standard deviation = 100 Random Numbers 0 95 0 91 0 87 0 83 0 79 0 75 0 71 0 67 0 63 0 59 0 55 0 51 0 47 0 43 0 39 0 30 25 20 15 10 5 0 35 Frequency Normal Distribution 15 RANDOM NUMBER GENERATION: EXAMPLE – A histogram of random numbers: exponential distribution, mean = 20 Exponential Distribution 30 20 10 Random Numbers 76 71 66 61 56 51 46 41 36 31 26 21 16 11 6 0 1 Frequency 40 16