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Chapter 7

Chapter 7. Section 7.1 – Discrete and Continuous Random Variables. Introduction. Sample spaces need not consist of numbers. When we toss four coins, we can record the outcomes as a string of heads and tails, such as HTTH.

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Chapter 7

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  1. Chapter 7 Section 7.1 – Discrete and Continuous Random Variables

  2. Introduction • Sample spaces need not consist of numbers. When we toss four coins, we can record the outcomes as a string of heads and tails, such as HTTH. • Recall from chapter 6 that a random variable is defined as a variable whose value is a numerical outcome of a random phenomenon. • In this section we will learn two ways of assigning probabilities to the values of a random variable. • Discrete • Continuous

  3. Discrete Random Variable • A discrete random variableX has a countable number of possible values. • The probability distribution of X lists the values xiand their probabilities pi: Value of X: x1x2x3 … Probability: p1p2p3 … • The probabilities pi must satisfy two requirements: • Every probability piis a number between 0 and 1. • The sum of the probabilities is 1. • To find the probability of any event, add the probabilities piof the particular values xithat make up the event.

  4. Example 7.1 - Getting Good Grades • See example 7.1 on p.392 • Probability histograms can be used to display probability distributions. • When using a histogram the height of each bar shows the probability of the outcome at its base. • Because the heights are probabilities, they add up to 1. • The bars are the same width.

  5. Example 7.2 – Tossing Coins • See example 7.2 on p.394-395

  6. Continuous Random Variable • A continuous random variableX takes all values in an interval of numbers. • The probability distribution of X is described by a density curve. • The probability of any event is the area under the density curve and above the values of X that make up the event.

  7. Example 7.3 – Random Numbers and the Uniform Distribution • See example 7.3 on p.398

  8. Continuous Random Variables • We assign probabilities directly to events – as areas under a density curve. Any density curve has area exactly 1 underneath it, corresponding to total probability 1. • All continuous probability distributions assign probability of 0 to every individual outcome. • Read p.399 for an explanation • We can ignore the distinction between < and when finding probabilities for continuous random variables. We can see why an outcome exactly to .8 should have probability of 0.

  9. Homework: p.396-405 #’s 2, 3, 6, 7, 10, 11, 14, 16, & 17

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