07_Discrete Probability Distributions Part 1

Rolling two dice probability mass function 0000 0028

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Rolling Two Dice (Probability Mass Function) 0.000 0.028 0.056 0.084 0.112 0.140 0.168 0.196 2 3 4 5 6 7 8 9 10 11 12 Sum of two dices Probability = = ) 7 ( X P = = ) 7 ( ) 7 ( X P F Rolling Two Dice (Cumulative Distribution) 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 2 3 4 5 6 7 8 9 10 11 12 Sum of two dices Probability

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22 More on random variables A random variable is not a fixed number, it can take on a range of values Information on how “large” the random variable is and how “spread out” or “risky” it is can be useful to a decision maker
23 Expected Value of a Random Variable The mean of a probability distribution A probability-weighted average of possible outcomes. E[X]: Expected value of x x: Values of the random variable P(x): Probability of the random variable taking on the value x. ) ( * ] [ x P x X E X = = μ

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24 he expected value is not the most likely outcome! Or even necessarily a possible outcome!) Does not always represent where a majority of value lies. Need to account for the spread of possible values. 50% of the time, the ball hooks 50% of the time, the ball slices On average, the ball should be always on green. Caveat of expected value
25 Variance and standard deviation of a random variable The variance of a random variable X is the weighted sum of its squared deviation from its mean Standard deviation : the square root of the variance E[X]: Expected value of x x: Values of the random variable P(x): Probability of the random variable taking on the value x. ) ( )] ( [ ] ) [( ] [ 2 2 2 x P X E x X E X Var X X - = - = = μ σ ] [ ] [ X Var X Stdev X = = σ

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26 An employee for a vending concession at Wells Fargo Center must choose between working the concession stand and receiving \$50 or walking around the stands selling hot dogs on a commission basis. Assume that the employee will make either \$90, \$70, \$45 or \$15 selling hot dogs with probabilities 0.1, 0.3, 0.4, and 0.2 respectively. Let the random variable X represent the profit made by the employee selling hot dogs. Define the probability mass function (pmf) for X. 90 0.1 Example: Selling hot dogs at Wells Fargo Center 70 0.3 45 0.4 15 0.2 x P(X=x)
27 1. Find the expected profit, E[X] when selling hot dogs.

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• Fall '12
• StephenD.Joyce
• Probability theory, discrete random variable

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