Chapter 16

# Chapter 16 - 500 may have a minor injury a Create a model...

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Chapter 16 Random Variables ActivStats: 16 Read: Chapter 16 A random variable = numerical description of outcomes of a random experiment Continuous – set of possible values is an interval Discrete - one can list all possible values Examples 1. X = sum of dots on two dice (discrete) 2. Y = GPA of a randomly chosen student (continuous) 3. W = lifetime of a light bulb (cont.) Probability Model for a Continuous Random Variable - a curve (Chapter 6) Probability Model for a Discrete Random Variable - a list of all possible values x and corresponding probabilities P(X = x) Model Parameters: Expected value : ) ( X E = μ the center of distribution = = = ) ( ) ( x X P x X E μ Variance : ) ( 2 X Var = σ measure of the spread = - = = ) ( ) ( ) ( 2 2 x X P x X Var μ σ Standard deviation : ) ( ) ( X Var X SD = = σ

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Example (Exercise 18) An insurance policy costs \$100 and will pay policyholder \$10,000 if they suffer a major injury or \$3,000 if they suffer a minor injury. The company estimates that each year 1 in every 2,000 policyholders may have a major injury, and 1 in

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Unformatted text preview: 500 may have a minor injury a) Create a model for X=profit on one policy b) What is the company expected profit on a policy? c) What is its standard deviation? a) b)– c) x 100-9,900-2,900 P(X=x) 0.9975 0.0005 0.0020 x P(X=x) xP(X=x) (x-μ ) (x - ) 2 P(X=x) 100 0.9975 99.75 11 120.70-9,900 0.0005-4.95-9989 49890.06-2,900 0.0020-5.8-2989 17868.24 Total 1 89 67879.00 E(X) = 89 Var(X) = 67879 SD(X) = √ 67879 = 260.54 Properties o E(X ± c) = E(X) ± c • Var(X ± c) = Var(X) o E(aX) = a E(X) • Var(aX) = a 2 Var(X) o E(X ± Y) = E(X) ± E(Y) o If X and Y are independent, then Var(X ± Y) = Var(X) + Var(Y) (always +) Example. Suppose that X and Y are independent and that E(X)=80, SD(X)=12 E(Y)=10, SD(Y)=3. Let W = X - 5Y + 20. Then: • E(W) = 80 –(5)(10) + 20 = 50 • Var(W) = Var(X - 5Y + 20) = Var(X - 5Y) = = Var(X) + Var( - 5Y) = Var(X) + (-5) 2 Var(Y) = = 12 2 + 25* 3 2 = 369, • SD(W) = 19.21...
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