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RANDOM VARIABLES

RANDOM VARIABLES. Random variables Probability distribution Random number generation Expected value Variance Probability distributions. RANDOM VARIABLES. Random variable: A variable whose numerical value is determined by the outcome of a random experiment Discrete random variable

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RANDOM VARIABLES

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  1. RANDOM VARIABLES • Random variables • Probability distribution • Random number generation • Expected value • Variance • Probability distributions

  2. 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

  3. 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.

  4. 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 0.75 0.5 Probability 0.25 0 HH HT TT Event

  5. 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

  6. CONTINUOUS PROBABILITY DSTRIBUTION f(x) z

  7. CONTINUOUS PROBABILITY DSTRIBUTION f(x) z

  8. 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.

  9. 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?

  10. REVISIT SIMPLE RANDOM SAMPLING • Plot a histogram with all the random numbers, and comment on the distribution of the random numbers.

  11. 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.

  12. 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

  13. RANDOM NUMBER GENERATION: EXAMPLE • A histogram of random numbers: uniform distribution, min = 500 and max = 800 Uniform Distribution 25 20 15 Frequency 10 5 0 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 Random Numbers

  14. RANDOM NUMBER GENERATION: EXAMPLE • A histogram of random numbers: triangular distribution, min = 3.2, mode = 4.2, and max = 5.2

  15. RANDOM NUMBER GENERATION: EXAMPLE • A histogram of random numbers: normal distribution, mean = 650 and standard deviation = 100

  16. RANDOM NUMBER GENERATION: EXAMPLE • A histogram of random numbers: exponential distribution, mean = 20

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