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Random Variables and Probability Distributions

Learn about discrete and continuous random variables, assigning probabilities to outcomes, probability distributions, and using histograms to display probabilities. Explore examples with tossing coins and random numbers. Understand how to assign probabilities to events with continuous variables.

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Random Variables and Probability Distributions

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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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