Question: [28 points] We’ve collected data on the percentage of the population of each county in California that is considered obese, using clinical measures, and that has been diagnosed with diabetes. We have constructed two random variables, Obesity and Diabetes, of the following forms (note: how the categories are constructed is irrelevant),

Obesity = 1 if percent obese is <18.6

2 if percent obese is 18.6 – 23.5

3 if percent obese is >23.5

Diabetes = 1 if percent w/ diabetes is <6.4

2 if percent w/ diabetes is 6.4 – 7.6

3 if percent w/ diabetes is >7.6

The following table reflects the joint probability function of our two variables:

a) [5 pts] What is the probability that Obesity = 2, and what is the probability that Diabetes = 3?

b) [5 pts] Calculate the conditional distribution of Diabetes given that Obesity = 1(i.e. fill in the table below).

c) [4 pts] Calculate E[Diabetes | Obesity=1].

d) [10 pts] Find the covariance and correlation of Diabetes and Obesity, and provide an interpretation of the sign of the statistic (explain the meaning of relationship between Diabetes and Obesity), show all your work.

e) [4 pts] Are Diabetes and Obesity independent? How do you know?