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Understanding Random Variables: Mean and Variance Calculation”

Learn about random variables, probability distribution, and the law of large numbers. Study how to calculate mean and variance of discrete random variables and distributions. Gain insights into combining independent random variables for sums and differences.

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Understanding Random Variables: Mean and Variance Calculation”

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  1. 5-Minute Check on section 7-2b Large Numbers The Law of ______ _________ says that as n increases x-bar  μ. Given X and Y are normally distributed with μx = 4, μy = 7, σx = 2, and σy = 3; find the following Mean of 2X + 3Y Variance of 2X + 3Y Mean of 3X – 5 Variance of 3X – 5 Mean of 4X – Y Variance of 4X – Y μ2X+3Y = 2μX + 3μY = 2(4) + 3(7) = 29 σ²2X+3Y = 2²σ²X + 3²σ²Y = 4(4) + 9(3) = 43 μ3X-5 = 3μX – 5 = 3(4) – 5 = 7 σ²3X-5 = 3²σ²X = 9(4) = 36 μ4X-Y = 4μX – μY = 4(4) – 7 = 9 σ²4X-Y = 4²σ²X + σ²Y = 4²(4) + (9) = 73 Click the mouse button or press the Space Bar to display the answers.

  2. Lesson 7 - R Review of Random Variables

  3. 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

  4. Vocabulary • Nothing

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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)

  14. 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 • Use your calculator to do the computations • Linear Combinations Rules • Adding a number changes mean, but not the variance • Multiplying a number changes mean and variance • Homework • pg 505-7; 7.53-62

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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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