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Chapter 7 Discrete Distributions Random Variable • A numerical variable whose value depends on the outcome of a chance experiment Two types: • Discrete – count of some random variable • Continuous – measure of some random variable Discrete Probability Distribution • Gives the values associated with each possible x value • Usually displayed in a table, but can be displayed with a histogram or formula Properties for a discrete probability distribution 1) For every possible x value, 0 < P(x) < 1. 2) For all values of x, S P(x) = 1. Suppose you toss 3 coins & record the number of heads. The random variable X defined as ... The number of heads tossed Create a probability distribution. X P(X) 0 .125 1 .375 2 .375 3 .125 Create a probability histogram. Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 P(X).02 .03 .09 4 5 6 7 ? .40 .16 .05 Why does this not start at zero? .25 P(x = 4) = P(x < 4) = .14 P(x < 4) = .39 P(x > 5) = .61 What is the probability that the student is registered for at least five courses? Formulas for mean & variance x x x 2 xi p i i x 2 p i Found on formula card! Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X).02 .03 .09 .25 .40 .16 .05 What is the mean and standard deviations of this distribution? 4.66 & = 1.2018 Here’s a game: If a player rolls two dice and gets aAsum of 2is or 12, fair game one where the cost to play EQUALS he wins $20.theIf he gets a expected value! 0 $5. 5The cost 20 7,X he wins to P(X) 7/9 1/6 1/18 roll the dice one time is $3. Is this game fair? NO, since = $1.944 which is less than it cost to play ($3). Linear function of a random variable The mean is changed If xbyis addition a random&variable and a and b aremultiplication! numerical constants, then the random variable y is defined by The standard deviation is y a bx ONLY changed by multiplication! • and y a bx a b x 2 2 2 2 y a bx b x or y b x Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed? = $622.40 & = $75.60 JustLinear add or subtract combinations the means! If y x 1 x 2 ... x n then y x 1 x 2 ... xn y 2 2 2 x 1 x 2 ... xn If independent, always add the variances! A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be mean SD MC 38 6 FR 30 7 If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test? 68 & = 9.2195