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HW- pgs. 475-476 (7.8-7.10) 7.1 Quiz THURSDAY Ch. 7 Test THURSDAY 12-22 www.westex.org HS, Teacher Website 12-13(=)-11 Warm up—AP Stats Welcome back to class James. To reacclimate you to AP Stats class I am giving you a chance to pick a number between 0 and 1. I will have a random number generator pick a number. What is the probability that James will pick the same number? Are you reacclimated James? Name _________________________ AP Stats 7 Random Variables 7.1 Day 2 Discrete & Continuous Random Variables Date _______ Objectives Define a density curve. Explain what is meant by a uniform distribution. Define a continuous random variable and a probability distribution for a continuous random variable. Continuous Random Variables When we use a table of __________ digits to select a digit between 0 and 9, the result is a 1 discrete random variable. The probability model assigns probability to each of the 10 10 possible outcomes. What if we wanted to choose a number at random between 0 and 1? To visualize this think of a spinner marked from 0 to 1. The sample space is now an entire ____________ of numbers: S = {all numbers x such that 0 X 1 } Can we assign the probability of getting a .238? How about the probability of getting between a .3 and a .4? We can’t add up individual value’s probabilities as we did with discrete random variables. Why? *** *** With continuous random variables the way to assign probabilities to events is by finding the _______ under a density curve, such as a normal curve or simple functions which we can use geometry to compute the areas. Any density curve has an area exactly _____ underneath it, corresponding to a total probability of 1. Take a look at example 7.3… The probability distribution of a continuous random variable assigns probabilities as area under a density curve. See figure 7.6. ***A continuous random variable X takes all values in an ____________ of numbers. The probability of any event is the _______ under the density curve and above the values of that make up the event*** See figure 7.6 above. REMEMBER---Every individual outcome has a probability of _____ for continuous probability distributions‼! You need an INTERVAL to assign positive probabilities. Since the probability of X = 0.7 is), it should make sense that the events X ≥ 0.7 and X > 0.7 have the same probability. With continuous random variables (but not ___________) it doesn’t make sense to make a distinction between ≥ and > when finding probabilities. ***Again, think of probabilities in continuous random variables as areas under a density curve. What is the area under a density curve above 0.7? What’s the width of 0.7? IT HAS NO WIDTH‼!! We need an interval in order to have a width so that we can find the AREA above the interval and below the density curve. Since there can’t be an area above a single point, such an even would have a probability of _____.*** Normal Distributions as Probability Distributions Normal curves are one example of density curves. (from section 2.2) Density curves describe assignments of probabilities (within intervals!) so Normal distributions are ______________ distributions. N(μ, σ) means that we are talking about a Normal distribution with __________ μ and standard deviation σ. In the language of random variables, if X has the N(μ, σ) distribution, then the _________________ variable: X Z is a standard Normal random variable having the distribution N(0, 1). YOU TRY: 1. Let the random variable X be a random number with the uniform density curve in Figure 7.5 (on previous pg.). Find the following probabilities. a. P(X < 0.49) b. P(X ≤ 0.49) c. P(X ≥ 0.27) d. P(0.27 < X < 1.27) e. P(0.1 ≤ X ≤ 0.2 or 0.8 ≤ X ≤ 0.9) f. The probability that X is not in the interval from 0.3 to 0.8 HW- pgs. 475-476 (7.8-7.10) 7.1 Quiz THURSDAY Ch. 7 Test THURSDAY 12-22 www.westex.org HS, Teacher Website g. P(X = 0.5)