invest_3ed.pdf

Another property of a random variable is its variance

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Another property of a random variable is its variance . This measures how variable the values of the random variable will be. For a discrete random variable we can again use a type of weighted average, based on the probabilities of each value and the squared distances between the possible values of the random variable and the expected value. V(X) = ¦ u ± values possible all value of y probabilit X E value ) ( )) ( ( 2 (x) Calculate the variance of the number of matches. Also take the square root to calculate the standard deviation SD(X). V(X) = SD(X) = We will interpret this standard deviation similarly to how we did in Investigation A: how far the outcomes tend to be from the expected value. Here we are talking in terms of the probability model; in Investigation A we were talking in terms of the historical data. (y) Confirm that the value of the standard deviation you calculating makes sense considering the possible outcomes for the random variable. Discussion : Notice that we have used two methods to answer questions about this random process: x Simulation running the process under identical conditions a large number of times and seeing how often different outcomes occur x Exact mathematical calculations using basic rules of probability and counting This approach of looking at the analysis using both simulation and exact approaches will be a theme in this course. We will also consider some approximate mathematical models as well. You should consider these multiple approaches as a way to assess the appropriateness of each method. You should also be aware of situations where one method may be preferable to another and why.

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Chance/Rossman, 2015 ISCAM III Investigation B 18 Practice Problem B.A Suppose three executives (Annette, Barb, and Carlos) drop their cell phones in an elevator and blindly pick them back up at random. (a) Write out the sample space using ABC notation for the outcomes. (b) Carry out the exact analysis to determine how the probability of at least one executive receiving his or her own phone. (c) Calculate the expected number of matches for 3 executives. How does this compare to the case with 4 mothers? (d) Use the Random Babies applet to check your results. Practice Problem B.B Reconsider the Random Babies. Now suppose there were 8 mothers involved in this random process. (a) Calculate the (exact) probability that all 8 mothers receive the correct baby. [ Hint : First determine how many possible outcomes there are for returning 8 babies to their mothers.] (b) Calculate the probability that exactly 7 mothers receive the correct baby. (c) Using the Random Babies applet, approximate the probability that at least one of the 8 mothers receives the correct baby. How does your approximation compare to the probability of this event with 4 mothers?
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