Solution Found!
Refer to Exercise 1. a. Find the conditional probability
Chapter 2, Problem 3E(choose chapter or problem)
Refer to Exercise 1.
a. Find the conditional probability mass function \(p_{Y \mid X}(y \mid 0)\).
b. Find the conditional probability mass function \(p_{X \mid Y}(x \mid 1)\).
c. Find the conditional expectation \(E(Y \mid X=0)\).
d. Find the conditional expectation \(E(X \mid Y=1)\).
Equation Transcription:
Text Transcription:
p_{Y|X}(y|0)
p_{X|Y}(x|1)
E(Y|X=0)
E(X|Y=1)
Questions & Answers
QUESTION:
Refer to Exercise 1.
a. Find the conditional probability mass function \(p_{Y \mid X}(y \mid 0)\).
b. Find the conditional probability mass function \(p_{X \mid Y}(x \mid 1)\).
c. Find the conditional expectation \(E(Y \mid X=0)\).
d. Find the conditional expectation \(E(X \mid Y=1)\).
Equation Transcription:
Text Transcription:
p_{Y|X}(y|0)
p_{X|Y}(x|1)
E(Y|X=0)
E(X|Y=1)
ANSWER:
Solution :
Step 1 of 4:
The number of days on which the ozone standard is exceeded is X and
The number of days on which the particulate matter standard is exceeded is Y.
Then the joint probability mass function of X and Y is given in the table below.
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 |
Our goal is:
a). Find the conditional probability mass function .
b). Find the conditional probability mass function .
c). Find the conditional expectation .
d). Find the conditional expectation .
a).
Now we have to find the conditional probability mass function .
We have a 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 |
Then the conditional probability mass function is
and
Then, we substitute and
values.
Therefore is 0.3846.
and
Then, we substitute and
values.
Therefore is 0.4230.
and
Then, we substitute and
values.
Therefore is 0.1923.
Hence the conditional probability mass function is
.