ementary probability theory) the probability that the sum of the two dice will be each ot 2312. Use these probabilities to find the (true expected value and standard deviation of the sum of the two dice, and compare with the (sample) mean in cell J4; add to the spreadsheet cal- culation of the (sample) standard deviation of the sum of the two dice and compare with your exact analytical result for the standard deviation. 2. In the simulation of throwing two dice in Section 3.2.1, derive (from el- Keep the number of throws at 50. 3. Prove rigorously, using probability theory and the definition of the ex- pected value of a random variable, that in Section 3.2.2, E(Y) Ja h(x)da Start by writing E(Y) E(b a)h(X)] (b- a)E[h(X)], then use the definition of the expected value of a function of a random variable, and finally remember that X is continuously uniformly distributed on la, b] so has density function f(x) 1/(b- a) for a S x
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ementary probability theory) the probability that the sum of the two dice will be each ot 2312. Use these probabilities to find the (true expected value and standard deviation of the sum of the two dice, and compare with the (sample) mean in cell J4; add to the spreadsheet cal- culation of the (sample) standard deviation of the sum of the two dice and compare with your exact analytical result for the standard deviation. 2. In the simulation of throwing two dice in Section 3.2.1, derive (from el- Keep the number of throws at 50. 3. Prove rigorously, using probability theory and the definition of the ex- pected value of a random variable, that in Section 3.2.2, E(Y) Ja h(x)da Start by writing E(Y) E(b a)h(X)] (b- a)E[h(X)], then use the definition of the expected value of a function of a random variable, and finally remember that X is continuously uniformly distributed on la, b] so has density function f(x) 1/(b- a) for a S x <b. 4. Use QRISK, or another Excel add-in for static spreadsheet simulation, to extend the example in Section 3.2.2 to 10,000 values of X,, rather than just the 50 values in Model.03.02.x1s in Figure 3.2. Compare your results to those in Mode1.03.02.xls as well as to the (almost) exact numerical integral. 5. In the Monte Carlo integration of Section 3.2.2, add to the spreadsheet calculation of the standard deviation of the 50 individual values, and use
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