In a certain community, levels of air pollution may exceed federal standards for ozone or for particulate matter on some days. For a particular summer season, let X be the number of days on which the ozone standard is exceeded and let Y be the number of days on which the particulate matter standard is exceeded. Assume that the joint probability mass function of X and Y is given in the following table:

Y 

X 
0 
1 
2 
0 
0.10 
0.11 
0.05 
1 
0.17 
0.23 
0.08 
2 
0.6 
014 
0.06 
a. Find P(X = 1 and Y = 0)
b. Find P(X > 1 and Y < 2).
c. Find P(X < 1).
d. Find P(Y≥ 1).
e. Find the probability that the standard for ozone is exceeded at least once.
f. Find the probability that the standard for particulate matter is never exceeded.
g. Find the probability that neither standard is ever exceeded.
Answer :
Step 1 of 8:
Given, in a certain community, levels of air pollution may exceed federal standards for ozone or for particulate matter on some days. The joint probability mass function of X and Y is in the below table.
Y 

X 
0 
1 
2 
0 
0.10 
0.11 
0.05 
1 
0.17 
0.23 
0.08 
2 
0.06 
0.14 
0.06 
Here, X be the number of days on which the ozone standard is exceeded.
Y be the number of days on which the particulate matter standard exceeded.
The joint probability mass function of two discrete random variables X and Y is defined as
Step 2 of 8:
a). To find P(X = 1 and Y = 0).
From the above table,
P(X = 1 and Y = 0) = 0.17.
Therefore,
P(X = 1 and Y = 0) = 0.17.
Step 3 of 8:
b). To find
Then,
=
= 0.17+0.23+0.06+0.14.
= 0.6
Therefore, = 0.6.
Step 4 of 8:
c). To find
Then,
= 0.10+0.11+0.05
= 0.26
Therefore, = 0.26.