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Unformatted text preview: Expected Value and Variance Problems: Example 1: How many licks does it take to get to the center of a tootsie roll pop? Given our distribution, what is the expected value and variance of the number of licks it takes to get to the center of a tootsie roll pop? licks owl person 1 person 2 silly person Suppose that it costs $ .40 for a tootsie roll pop. Suppose further that you want to make sure that you get 500 licks of a tootsie roll pop, what would your expected cost be? Example 2: How much wood could a woodchuck chuck if a woodchuck could chuck wood? We have the following distribution, find the expected value and variance of the amount of wood a woodchuck could chuck in a day (measured in butt cords). amount of wood probability woodchuck 1 woodchuck 2 woodchuck 3 woodchuck 4 Suppose we had a family of 7 woodchucks, what is the expected value and variance of the amount of wood that the woodchuck family could chuck in a day? 153 272 573 1245 0.15 0.2 0.23 0.42 probability 3 100 0.001 0.55 200 0.4489999 427 0.000001 Peter Piper picked a peck of pickled peppers. If Peter Piper could pick the following number of pecks of peppers in a day, what is the expected value and variance of the number of pecks of pickled peppers that Peter Piper could pick in a day? # of Pecks probability 20 50 120 175 200 Every week Peter goes to the market on Saturday (he does not pick peppers on Saturday). He sells all of his pecks of peppers on Saturdays. If he gets $ .35 for a peck of peppers, what is the expected value and variance of the amount of money he will earn? Sally sells seashells by the seashore. Suppose on a given day she sells the following amount of shells with given probabilities: # of Shells probability 1 2 3 4 5 Suppose Sally sells her seashells for $2 a shell, how much money should she expect to get in a day? 0.25 0.15 0.3 0.2 0.1 0.01 0.25 0.35 0.2 0.19 Further, let us suppose that Sally's cost function is Y=.4*abs(X1.5), where abs represents absolute value, and X represents number of shells. First calculate the expected value and variance of Y, then check your work by finding the pmf of Y and recalculating the expected value and variance of Y. What do you notice about this problem? Conclusions? 6. The PMF of a discrete random variable X is described below. x -2 -1 0 0.04 1 0.19 2 0.11 3 0.15 pX(x) 0.22 0.29 (a) What is the probability X is between -0.8 and 2.2? (b) Given X is at least 0, what is the probability it is at least 1? (c) Find the expected value and variance of X. (d) Let Y = 2X - 1. Find the PMF of Y . (e) Let Z = X*X. Find the PMF of Z. (f) What is special about the function f(x) = 2x-1, compared to g(x) = x*x, for Our values of x, that makes part (d) easier than part (e)? 8. Let X be a random variable with PMF defined as p X ( x) = k (5 - x) 0 for x=0, 1, 2, 3, 4 otherwise (a) Find the value of k that makes pX(x) a legitimate PMF. (b) What is the probability that X is between 1 and 3, inclusive? (c) If X is not 0, what is the probability X is less than 3? (d) Let Z = 3X + 1. Find the expected value and variance of Z. New Example: p X ( x) = | x-2| 10 0 for x= -2, -1, 0, 1 otherwise What is the expected value of x? What is the variance of x? Write down the pmf table of x. Write down the pmf table for y, when y = x 2 . Write down the pmf table of z, when z = |x|. What are the expected values and variances of y and z? ...
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- Spring '08