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BBA182 Applied Statistics Week 6 (2) Random variables – discrete random variables Dr Susanne Hansen Saral Email: susanne.saral@okan.edu.tr

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Random Variables Represent possible numerical values from a random experiments. Which outcome will occur, is not known, therefore the word “random”.

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Discrete random variable A discrete random variable is a possible outcome from a random experiment. It takes on countable values, integers. Examples of discrete random variables: Number of cars crossing the Bosphorus Bridge every day Number of journal subscriptions Number of visits on a given homepage per day We can calculate the exact probability of a discrete random variable:

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Discrete random variable We can calculate the exact probability of a discrete random variable. Example: The probability of students coming late to class today out of all students registered Total students registered: 45 Students late for the class: 15 P(students late for the class) = = .33 or 33 %

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Continuous random variable A random variable that has an unlimited set of values. Therefore called continuous random variable Continuous random variables are common in business applications for modeling physical quantities such as height, volume and weight, and monetary quantities such as profits, revenues and expenses. Examples: The weight of cereal boxes filled by a filling machine in grams Air temperature on a given summer day in degrees Celsius Height of a building in meters Annual profits in $ of 10 Turkish companies

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Continuous random variable A continuous random variable has an unlimited set of values. The probability of a continuous variable is calculated in an interval (ex.: 5 -10), because the probability of a specific continuous random variable is close to 0. This would not provide useful information. Example: The time it takes for each of 110 employees in a factory to assemble a toaster:

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Time, in seconds, it takes 110 employees to assemble a toaster 271 236 294 252 254 263 266 222 262 278 288 262 237 247 282 224 263 267 254 271 278 263 262 288 247 252 264 263 247 225 281 279 238 252 242 248 263 255 294 268 255 272 271 291 263 242 288 252 226 263 269 227 273 281 267 263 244 249 252 256 263 252 261 245 252 294 288 245 251 269 256 264 252 232 275 284 252 263 274 252 252 256 254 269 234 285 275 263 263 246 294 252 231 265 269 235 275 288 294 263 247 252 269 261 266 269 236 276 248 299

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Continuous random variable The probability of a continuous variable is calculated in an interval (5 - 10), because the probability of a specific continuous random variable is close to 0. Example: The time it takes for each of 110 employees in a factory to assemble a toaster: n = 110 1 employee assembles the toaster is 222 seconds out of 110 P(employee assembles the toaster in 222 seconds out of all 110 employees) = = 0.009 or 0.9 % this provides little useful information

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Employee assembly time in seconds Completion time (in seconds) Frequency Relative frequency % 220 – 229 5 4.5 230 – 239 8 7.3 240 – 249 13 11.8 250 – 259 22 20.0 260 – 269 32 29.1 270 – 279 13 11.8 280 – 289 10 9.1 290 – 300 7 6.4 Total 110 100 %

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Probability Models For both discrete and continuous variables, the collection of all possible outcomes (sample space) and probabilities associated with them is called the probability model. For a discrete random variable, we can list the probability of all possible values in a table. For example, to model the possible outcomes of a dice, we let X be the random variable called the “number showing on the face of the dice”. The probability model for X is therefore: 1/6 if x = 1, 2, 3, 4, 5, or 6 P(X = x) = 0 otherwise

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Probability Model, also Probability Distributions Function, P(x) for Discrete Random Variables 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25

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Probability Distributions Function, P(x) for Discrete Random Variables (example) Sales of sandwiches in a sandwich shop: Let, the random variable X, represent the number of sandwiches sold within the time period of 14:00 - 16:00 hours in one given day. The probability distribution function, P(x) of sales is given by the table here below:

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Graphical illustration of the probability distribution of sandwich sales between 14:00 -16:00 hours

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Requirements for a probability distribution of a discrete random variable

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Cumulative Probability Function F(x0) = P(x) = 1;

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Cumulative Probability Function, F()

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Graphical illustration of P(x) F(x0) = P(x) = 1;

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Graphical illustration of F(x0) Cumulative probability distribution, Ogive

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Cumulative Probability Function, F(x0) Practical application

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Cumulative Probability Function, F(x0) Practical application: Car dealer

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Cumulative Probability Function, F(x0) Practical application

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Cumulative Probability Function, F(x0) Practical application

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The number of computers sold per day at Dan’s Computer World is defined by the probability distribution above: a) Calculate the cumulative probability distribution b) What are the following probabilities? P(3 P(x P(x P(2

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Cumulative probability - solution The number of computers sold per day at Dan’s Computer World is defined by the following probability distribution: P(3 P(x P(x P(2

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Exercise In a geography exam the grade students obtained is the random variable X. It has been found that students have the following probabilities of getting a specific grade: A: .18 D: .07 B: .32 E: .03 C: .25 F: .15 Based on this, calculate the following: The cumulative probability distribution of X, F(x) The probability of getting a higher grade than B The probability of getting a lower grade than C The probability of getting a grade higher than D The probability of getting a lower grade than B

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Cumulative probabilities - exercise Based on this, calculate the following: The cumulative probability distribution of X, F(x0) The probability of getting a higher grade than B, P(x > B) The probability of getting a lower grade than C, P(x < C) The probability of getting a grade higher than D P(x > D) The probability of getting a lower grade than B P(x < B)

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Properties of Discrete Random Variables The measurements of central tendency and variation for discrete random variables: Expected value E[X] of a discrete random variable - expectations Expected Variance of a discrete random variable Expected Standard deviation of a discrete random variable Why do we refer to expected value?

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Expectations Of course, we cannot predict exactly which number will occur when we roll a dice, but we can say what we expect to happen on average, in the long run (therefore the name expectation) The expected value of rolling a dice infinitively is a parameter of the probability model. In fact, it is the mean, We’ll write it as: E[X], for Expected value The expected value is not an average of data values, but calculated from the probability distribution of rolling one dice infinitively.

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Expected Value of a discrete random variable X: Example: Toss 2 coins, random variable, X = # of heads, (TT, HT,TH,HH) compute the expected value of X:

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Properties of Discrete Random Variables Expected Value (or mean) of a discrete random variable X: We weigh the possible outcomes by the probabilities of their occurrence: E(x) = (0 x .25) + (1 x .50) + (2 x .25) = 1.0

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Expected value So, the expected value, E[X], of a discrete random variable is found by multiplying each possible value of the random variable by the probability that it occurs and then summing all the products: The expected value of tossing two coins simultaneously is therefore:

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Concept of expected value of a random variable A review of university textbooks reveals that 81 % of the pages have no mistakes, 17 % of the pages have one mistake and 2% have two mistakes. We use the random variable X to denote the number of mistakes on a page chosen at random from a textbook with possible values, x, of 0, 1 and 2 mistakes. With a probability distribution of : P(0) = .81 P(1) = .17 P(2) = .02 How do we calculate the expected value(average) of mistakes per page?

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Expected value – calculation (example) Find the expected mean number of mistakes on pages: = (0)(.81)+(1)(.17)+(2)(.02) =.21 From this result we can conclude that over a large number of pages, the expectation would be to find an average of 21 % mistakes per page in business textbooks.

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Probability distribution of mistakes in textbooks

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Exercise A lottery offers 500 tickets for $ 3 each. If the biggest prize is $ 250 and 4 second prizes are $ 50 each : a) What are the possible outcomes? b) What is the expected value, E[X], of a single ticket? c) Now, include the cost of the ticket you bought. What is the expected value now? d) Knowing the value calculated in part b) does it make sense to buy a lottery ticket? e) What is the expected value the lottery company can expect to gain from the lottery sale?

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Exercise A lottery offers 500 tickets for $ 3 each. If the biggest prize is $ 250 and 4 second prizes are $ 50 each. a) What are the possible outcomes? Winning the large prize of $ 250, 1/500, winning one of the 4 prizes of $ 50, 4/500 and winning nothing, $ 0, 495/500 b) What is the expected value, E[X], of a single ticket? E[X] = $ 250 x (1/500) + $ 50x (4/500) + $ 0 x (495/500) = $ 0.50 +$ 0.40 +$ 0.00 = $ 0.90 c) Now, include the cost of the ticket you bought. What is the expected value now? E[X] = $ 0.90 - $ 3.00 = $ - 2.10 d) Knowing the value calculated in part b) does it make sense to buy a lottery ticket?

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Exercise Although no single person will lose $ 2.10, because they either lose $3 or win $ 250 or $ 50. $ 2.10 is the amount in average that the lottery organization gains per ticket. The lottery can therefore expect to make 500 x $ 2.10 = $ 1050 by selling 500 lottery tickets.