Refer to Exercise 4.

a. Find the conditional probability mass function PY|X(y|1).

b. Find the conditional probability mass function PX|Y(x|2).

c. Find the conditional expectation E(Y | X = 1).

d. Find the conditional expectation E(X | Y = 2).

REFERENCE EXERCISE 4: In a piston assembly, the specifications for the clearance between piston rings and the cylinder wall are very tight. In a lot of assemblies, let X be the number with too little clearance and let Y be the number with too much clearance. The joint probability mass function of X and Y is given in the table below:

a. Find the marginal probability mass function of X.

b. Find the marginal probability mass function of Y.

c. Are X and Y independent? Explain.

d. Find μX and μY.

e. Find σX and σY.

f. Find Cov(X, Y).

g. Find ρ(X, Y).

Step 1 of 5:

Here,it is given that X be the number with too little clearance and Y be the number with too much clearance.

Also the joint probability mass function of X and Y is given as

Y | ||||

X |
0 |
1 |
2 |
3 |

0 |
0.15 |
0.12 |
0.11 |
0.10 |

1 |
0.09 |
0.07 |
0.05 |
0.04 |

2 |
0.06 |
0.05 |
0.04 |
0.02 |

3 |
0.04 |
0.03 |
0.02 |
0.01 |

Using these,we have to find the required probabilities and expectations.