1. Consider the following discrete probability distribution function for the variable X:
|
X
|
10
|
12
|
14
|
16
|
18
|
|
f(x)
|
.02
|
.7
|
.05
|
.2
|
.03
|
a) Is this a proper probability mass function? How do you know?
b) Find the mean, variance, and standard deviation of X, showing your work.
c) What is the probability that any draw from this distribution, say X, is less than 16 [i.e. find P(X<16)]?
2. Consider the following discrete probability mass function for the variables Health Insurance and Income. Health Insurance indicates whether each individual possesses health insurance coverage and, if so, what type, with
and Income measures gross income class, with
The joint pmf is given as:
|
HI
|
Inc
|
|
1
|
2
|
3
|
|
1
|
.03
|
.13
|
.44
|
|
2
|
.01
|
.01
|
.03
|
|
3
|
.07
|
.03
|
.08
|
|
4
|
.10
|
.06
|
.01
|
a) Is this a proper probability mass function? How do you know?
b) What is the probability that an individual drawn from this distribution holds public health insurance? What type of probability is this (e.g. joint, marginal, or conditional)?
c) Construct the conditional probability distribution (pmf) of Income given that an individual is covered by a private, employer sponsored health insurance plan; i.e., construct the table or graph that shows, 
.
d) What is the probability that an individual earns $30,000 or more per year given that they are insured by at least some form of insurance (i.e. not uninsured)?
e) Calculate the mean income class for those individuals with private, employer sponsored insurance. Is this greater than or less than the mean income class for those with private, individually purchased insurance? Any thoughts regarding the comparison?
f) Find the covariance between HI and Inc and provide an interpretation of the sign of the statistic.
g) Are HI and Inc independent? How do you know?