AP Statistics Chapter 6 - Random Variables
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1 AP Statistics Chapter 6 - Random 6.1 Discrete and Continuous Random Objective: Recognize and define discrete random variables, and construct a probability distribution table and a probability histogram for the random variable. Recognize and define a continuous random variable, and determine probabilities of events as areas under density curves. Calculate the mean of a discrete random variable. Interpret the mean of a random variable in context. Calculate the standard deviation of a discrete random variable. Interpret the standard deviation of a random variable in context. Given a normal random variable, use the standard normal table or a graphing calculator to find probabilities of events as areas under the standard normal distribution curve Read page probability model random variable probability distribution Discrete Random A probability model describes the possible outcomes of a chance process and the likelihood that those outcomes will occur. A numerical variable that describes the outcomes of a chance process is called a random variable. The probability model for a random variable is its probability distribution The probability distribution of a random variable gives its possible values and their probabilities. There are two main types of random variables: discrete and continuous. If we can find a way to list all possible outcomes for a random variable and assign probabilities to each one, we have a discrete random variable discrete random variable Do #1,35,7 1
2 Read pages Mean (Expected Value of a Discrete Random Variable Mean When analyzing discrete random variables, we ll follow the same strategy we used with quantitative data describe the shape, center, and spread, and identify any outliers. The mean of any discrete random variable is an average of the possible outcomes, with each outcome weighted by its probability. Definition: Suppose that is a discrete random variable whose probability distribution is Value: x 1 x x 3 Probability: p 1 p p 3 To find the mean (expected value of, multiply each possible value by its probability, then add all the products: x E ( x1 p1 x p x3 p3... xi pi Do #11,13 Read page Standard Deviation and Variance of a Discrete Random Variable Since we use the mean as the measure of center for a discrete random variable, we ll use the standard deviation as our measure of spread. The definition of the variance of a random variable is similar to the definition of the variance for a set of quantitative data. Variance and Standard Deviation Definition: Suppose that is a discrete random variable whose probability distribution is Value: x 1 x x 3 Probability: p 1 p p 3 and that µ is the mean of. The variance of is Var ( ( x ( x i 1 p i p ( x 1 p ( x 3 p... To get the standard deviation of a random variable, take the square root of the variance. 3 Do #15,17 Read and Do Technology Corner page 348 Do #19 page 355 (using calculator
3 Read pages Continuous Random Discrete random variables commonly arise from situations that involve counting something. Situations that involve measuring something often result in a continuous random variable Definition: A continuous random variable takes on all values in an interval of numbers. The probability distribution of is described by a density curve. The probability of any event is the area under the density curve and above the values of that make up the event. The probability model of a discrete random variable assigns a probability between 0 and 1 to each possible value of. A continuous random variable Y has infinitely many possible values. All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability. Do #1, 3, 5 Do Review Problems #
4 6. Transforming and Combining Random Read pages Remember: Objectives: Describe the effects of transforming a random variable by adding or subtracting a constant and multiplying or dividing by a constant. Find the mean and standard deviation of the sum or difference of independent random variables. Determine whether two random variables are independent. Find probabilities involving the sum or difference of independent Normal random variables. In Chapter, we studied the effects of linear transformations on the shape, center, and spread of a distribution of data. Recall: 1. Adding (or subtracting a constant, a, to each observation: Adds a to measures of center and location. Does not change the shape or measures of spread.. Multiplying (or dividing each observation by a constant, b: Multiplies (divides measures of center and location by b. Multiplies (divides measures of spread by b. Does not change the shape of the distribution. Linear Transformations How does multiplying or dividing by a constant affect a random variable? How does adding or subtracting a constant affect a random variable? Effect of a Linear Transformations on the Mean and Standard Deviations Do #35, 37,39,40,41,43,45 4
5 Read pages Mean of the Sum of Random Combining Random Independent Random Definition: If knowing whether any event involving alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then and Y are independent random variables. Probability models often assume independence when the random variables describe outcomes that appear unrelated to each other. You should always ask whether the assumption of independence seems reasonable. In our investigation, it is reasonable to assume and Y are independent since the siblings operate their tours in different parts of the country. Variance of the Sum of Random Remember that you can add variances only if the two random variables are independent, and that you can NEVER add standard deviations! Mean of the Difference of Random For any two random variables and Y, if D = - Y, then the expected value of D is E( D D In general, the mean of the difference of several random variables is the difference of their means. The order of subtraction is important! x Y Variance of the Difference of Random For any two independent random variables and Y, if D = - Y, then the variance of D is D In general, the variance of the difference of two independent random variables is the sum of their variances. Y Read pages Combining Normal Random An important fact about Normal random variables is that any sum or difference of independent Normal random variables is also Normally distributed. Do #61,63 Do Review Problems #67,68 5
6 6
Chapter 6: Random Variables
Chapter 6: Random Variables Section 6.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 6.3 Binomial and
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