Problem set 1

Consider the above random variable, \(X\), with its associated probability distribution:

1. Draw the probability distribution function and the cumulative distribution function.

2. What is the expected value of X? That is, what is \(E[X]\)?

3. What is the variance of X?

Stock and Watson 2.6, 2.10, (note that I originally assigned 2.18, 3.16, but the content is covered in Week 3 - you can submit these now or with PS2)

For a randomly selected county in the United States, let \(X\) represent the proportion of adults over age 65 who are employed (the elderly employment rate). Then, \(X\) is restricted to a value between zero and one. Suppose that the cumulative distribution function for \(X\) is given by \(F(x) = 3x^2 - 2x^3\) for \(0 \leq x \leq 1\).

1. What is the probability that the elderly employment rate is at least 0.5 (50%)?

2. What is the probability that the elderly employment rate is between 0.4 (40%) and 0.6 (60%)?