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Lesson 7 - R

Lesson 7 - R. Review of Random Variables. Objectives. Define what is meant by a random variable Define a discrete random variable Define a continuous random variable Explain what is meant by the probability distribution for a random variable

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Lesson 7 - R

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  1. Lesson 7 - R Review of Random Variables

  2. Objectives • Define what is meant by a random variable • Define a discrete random variable • Define a continuous random variable • Explain what is meant by the probability distribution for a random variable • Explain what is meant by the law of large numbers • Calculate the mean and variance of a discrete random variable • Calculate the mean and variance of distributions formed by combining two random variables

  3. Vocabulary • Nothing

  4. AP Outline Fit: III. Anticipating Patterns: Exploring random phenomena using probability and simulation (20%–30%) A. Probability 2. “Law of Large Numbers” concept 4. Discrete random variables and their probability distributions, including binomial and geometric 6. Mean (expected value) and standard deviation of a random variable, and linear transformation of a random variable B. Combining independent random variables 1. Notion of independence versus dependence 2. Mean and standard deviation for sums and differences of independent random variables

  5. What we Learned • Random Variables • Recognize and define a discrete random variable, 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 • 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

  6. What we Learned • Means and Variances of Random Variables • Calculate the mean and variance of a discrete random variable. Find the expected payout in a raffle or similar game of chance • Use simulation methods and the law of large numbers to approximate the mean of a distribution • Use rules for means and rules for variances to solve problems involving sums, differences, and linear combinations of random variables

  7. Using your TI-83 calculator We can use 1-Var-Stats to calculate the mean and standard deviation of a discrete random variable given it’s outcomes and probability • Type in outcomes in L1 • Type in corresponding probabilities in L2 • Use 1-Var-Stats L1, L2 to get statistics • Notes: • Discrete Random Variables have countable (finite) values • Continuous Random Variables have an interval of values (infinite) • Ranges of Random Variables are determined by minimum or maximum values that they can take on

  8. Discrete Random Variable - Mean The mean, or expected value [E(x)], of a discrete random variable is given by the formula μx = ∑ [x ∙P(x)] where x is the value of the random variable and P(x) is the probability of observing x (multiply them together and add all of them up) Mean of a Discrete Random Variable Interpretation: If we run an experiment over and over again, the law of large numbers helps us conclude that the difference between x and ux gets closer to 0 as n (number of repetitions) increases

  9. Discrete Random Variable - Variance Variance and Standard Deviation of a Discrete RV: The variance of a discrete random variable is given by: σ2x = ∑ [(x – μx)2 ∙ P(x)] = ∑[x2 ∙ P(x)] – μ2x and standard deviation is √σ2 Note: round the mean, variance and standard deviation to one more decimal place than the values of the random variable

  10. Probability Laws • Law of Large Numbers – True • Sample mean, x, approaches population mean, μ, as sample size increases • Law of Small Numbers – False • No such thing • Random behavior in short term does not mimic long-term behavior • Law of Averages – Bad Statistics • Eventually everything evens out • Each trial is independent

  11. Random Variables and Probability • Area under the probability density function (PDF) curve between the values of the random variable determine the probability • Without calculus the only continuous random variable PDFs we can use are • Normal (calculator and tables) • Uniform (always forms a rectangle) • Piece-wise linear (other known geometric areas) • Discrete PDFs are calculated by summing up the given (or calculated) probabilities

  12. Means and Variances • Rules for Means • Means follow the rules for linear combinations (from Algebra) • When you linearly combine two or more (rules give only the 2 case example) random variables, you combine their means in the same manner • E(a + X + bY) = a + E(X) + bE(Y) • Rules for Variances • Adding a number to a random variable does not change its variance • Multiply a random variable by a number changes the variance by the square of that number • V(a + X + bY) = V(X) + b²V(Y) • When you combine random variables, you always add the variances • V(X - Y) = V(X) + V(Y) = V(X + Y)

  13. Summary and Homework • Summary • Random Variables (RV) • Discrete RV – finite outcomes • Continuous RV – an interval outcomes (infinite) • Mean, Variance and Standard Deviation of RV • Discrete RV – know the formulas • Continuous RV – memorize for each distribution we study • Linear Combinations Rules • Homework • pg 505-7; 7.53-62

  14. Problem 1a The random variable X represents the number of people that you have to wait behind in line when you go to the post office to buy stamps at lunch time. The probability distribution of X is provided below: X = 0 1 2 3 Probability = .1 .5 .3 .1 • Find the mean number of people that will be in front of you in the stamp line. Use the definition and show work. Mean: ∑ [x ∙P(x)] = (.1)(0) + (.5)(1) + (.3)(2) + (.1)(3) = 0 + .5 + .6 + .3 = 1.4

  15. Problem 1b The random variable X represents the number of people that you have to wait behind in line when you go to the post office to buy stamps at lunch time. The probability distribution of X is provided below: X = 0 1 2 3 Probability = .1 .5 .3 .1 (b) Find the standard deviation for the number of people in the line in front of you. Use the definition and show work. Var: ∑[x2 ∙ P(x)] – μ2x = ∑ [x2 ∙ P(x)] – μx2 = (0 + .5 + .3(4) + .1(9) ) – 1.96) = 2.6 – 1.96 = 0.64 St Dev = 0.8

  16. Problem 2 From the previous problem, let f(X) = 2X + 0.5 represent the amount of time (in minutes) required for the clerks to process X people. Show your work and use the shortcut methods (not the definitions) to find: • The mean number of minutes that you will have to wait. • The standard deviation of the number of minutes you will have to wait. μX = 1.4 so μf(X) = 0.5 + 2 μX = 0.5 + 2(1.4) = 3.3 minutes σX = 0.8 so σ²f(X) = 2² σ²X = 4 (0.8)² = 2.56 minutesσf(X) = 2.56 = 1.6 minutes

  17. Problem 3 While you are at the post office you also need to pick up a package. The random variable Y represents the number of people you have to wait behind in the pickup line. The probability distribution of Y is provided below: Y = 0 1 2 Probability = .2 .3 .5 • Use your calculator to find the mean number of people that will be in front of you in this line • Use your calculator to find the standard deviation of the number of people in this line Mean: ∑ [x ∙P(x)] = 1.3 Var: ∑[x2 ∙ P(x)] – μ2x = 0.781

  18. Problem 4 Suppose that the numbers of people in the two lines are independent of each other. Let Z = X + Y represent the total number of people you will have to wait behind at the post office. Use the rules we discussed in class to find: • The mean or expected value of Z. • The standard deviation of Z. E(Z) = E(X) + E(Y) = 1.4 + 1.3 = 2.7 V(Z) = V(X) + V(Y) = 0.8² + 0.781² = 1.25 σ(Z) = 1.25 = 1.118

  19. Problem 5 The weight of eggs produced by a certain breed of hen is normally distributed with mean of μ = 65 grams and standard deviation of σ = 5 grams. What is the probability that the weight of a dozen (12) randomly selected eggs is between 750 grams and 800 grams? E(D) = μD = μE1 + μE2 + … + μE12 = 12 μE = 12  65 = 780 grams V(D) = σ²D = σ²E1 + σ²E2 + σ²E3 … + σ²E12 = 12 σ²E = 12  5² = 300 grams σD = 300 = 17.32 750 800 780 normalcdf(750, 800, 780, 17.32) = 83.43%

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