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A. P. STATISTICS LESSON 7 – 1 ( DAY 1 ) DISCRETE AND CONTINUOUS RANDOM VARIABLES ESSENTIAL QUESTION: What are discrete and continuous random variables, and how are they used to determine probabilities? • To define discrete random variables. • To use discrete random variables to solve probability problems. • To define continuous random variables. • To use continuous random variables to solve probability problems. 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. • Random Variable – A random variable is a variable whose value is a numerical outcome of a random phenomenon. Discrete and Continues Random Variables Discrete and Continuous are two different ways of assigning probabilities. Discrete Random Variables A discrete random variable X has a countable number of possible values. The probability of X lists the values and their probabilities: Values of X: x1 , x2 , x3 , …. Xk Probability: p1 , p2 , p3 , …pk The probabilities pi must satisfy two requirements: 1. Every probability pi is a number between 0 and 1. 2. p1 + p2 + …. + pk = 1 Find the probability of any event by adding the probabilities pi of the particular values xi that make up the event. Example 7.1 Getting Good Grades Page 392 • 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. Example 7.2 Tossing Coins Page 394 Two assumptions are made: 1. The coin is balanced, so each toss is equally likely to give H or T. 2. The coin has no memory, so tosses are independent. Continuous Random Variables When using the table of random digits to select a digit between 0 and 9, the result is a discrete random variable. The probability model assigns probability 1/10 to each of the 10 possible outcomes. Continuous random variable is a number at random from 0 to 1. Continuous Random Variables A continuous random variable X 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. Discrete and Continuous Random Variables Exercises Page 395 - 397 • • • • • 7.1 Roll of the Die 7.2 Three Children 7.3 Social Class in England 7.4 Housing in San Jose, I 7.5 Housing in San Jose, II Example 7.3 Random Numbers and the Uniform Distribution Page 398 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. We can see why an outcome exactly to .8 should have probability of 0. Normal Distributions as Probability Distributions The density curves that are most familiar to us is the normal curves. Normal distributions are probability distributions. Recall that N(μ, σ) is our shorthand notation for the normal distribution having mean μ and standard deviation σ. Z=X-μ σ