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Understanding Discrete Probability Distributions and Random Variables

This chapter delves into discrete distributions, focusing on random variables whose values depend on chance experiments. It defines discrete variables, providing properties and examples, such as tossing coins and student course registrations. The chapter explains how to create probability distributions and histograms, find means and variances, and evaluate scenarios of chance, like rolling dice. Moreover, it discusses the linear transformation of random variables and the implications on means and standard deviations, equipping readers with a comprehensive understanding of discrete probability concepts.

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Understanding Discrete Probability Distributions and Random Variables

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  1. Chapter 7 Discrete Distributions

  2. Random Variable - • A numerical variable whose value depends on the outcome of a chance experiment

  3. Two types: • Discrete – count of some random variable • Continuous – measure of some random variable

  4. 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

  5. Properties for a discrete probability distribution • For every possible x value, 0 < P(x) < 1. 2) For all values of x, S P(x) = 1.

  6. Suppose you toss 3 coins & record the number of heads. The random variable X defined as ... Create a probability distribution. Create a probability histogram. The number of heads tossed X 0 1 2 3 P(X) .125 .375 .375 .125

  7. 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 ? .40 .16 .05 P(x = 4) = P(x < 4) = P(x < 4) = What is the probability that the student is registered for at least five courses? Why does this not start at zero? .25 .14 P(x > 5) = .61 .39

  8. Formulas for mean & variance Found on formula card!

  9. 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? m = 4.66 & s = 1.2018

  10. X 0 5 20 P(X) 7/9 1/6 1/18 Here’s a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair? A fair game is one where the cost to play EQUALS the expected value! NO, since m = $1.944 which is less than it cost to play ($3).

  11. If x is a random variable and a and b are numerical constants, then the random variable y is defined by and Linear function of a random variable The mean is changed by addition & multiplication! The standard deviation is ONLY changed by multiplication!

  12. 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? m = $622.40 & s = $75.60

  13. Linear combinations Just add or subtract the means! If independent, always add the variances!

  14. 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? m = 68 & s = 9.2195

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