Question

2. Expected Values:
a) What is the formula for Expect[X] for some data set X, when X is continuous and when X is discrete?
b) Must an expected value of a data set be an element of that set? Why or why not?
c) How is the expected value similar to or different from other measures of center such as the median or mode?
d) What is the difference between theoretical expected values and experimental (sample) means?
i. When is it a safe bet to say that an experimental (sample) mean will be close in value to the theoretical expected value?

e) What are some properties of the expected value? (Know how to show the following for continuous or discrete random
variables.)

i. Expect[X+c] = Expect[X] + c
ii. Expect[c X] = c Expect[X]
ii. Expect[X Y] = Expect[X] Expect[Y] (assuming independence of X and Y)

f) How do you calculate Expect[g[x]] for some function g[x] and data set X?
i. How does this relate to the computation of the variance of a data set?

Answer 1