Hints for Homework Assignment 8

General Remarks

For normal approximation problems, I recommend the formula for P(Sn < = x) (which was used in all class examples and is given on the handout rather than the one in the book on p. 196). (The two formulas are equivalent, so either one can be used but the one in the book is more cumbersome to apply.)

Specific Comments

• Problem 1: This is a routine normal approximation problem that can be done by hand, without calculator.
• Problem 2: An easy exercise in working with the normal distribution. No approximation needed here since the given distribution is already normal.
• Problem 3: Another routine application of normal approximation.
• Problem 4: This problem is more interesting. Part (a) is nearly trivial (and has nothing to do with normal approximation). Part (b) is more interesting and requires defining appropriate i.i.d. variables and applying normal approximation to the sum of these random variables. The two strategies can be characterized as putting all eggs in one basket (strategy (a)) and spreading out the risk evenly (strategy (b)). The question which strategy is more likely to result in a profit of $8000, is of considerable practical interest.
• Problem 5: An example of applying normal approximation to averages of i.i.d. random variables. The results should be in line with the general behavior of such averages: namely that as n gets larger and larger, the averages A_n tend to be more and more concentrated near the expected value of X1. (This phenomenon is called the "law of large numbers".)
• Problem 6: The key is to cast the problem in terms of sums of i.i.d. random variables, so that normal approximation can be applied. How these random variables should be defined is less obvious here than in most other applications of normal approximation. A correct and precise definition of the r.v.'s is essential.

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