Answer:
a) Joint ptobability distribution

b) P(X<= 1 and Y <= 1) = P(X<= 1) * P(Y<=1) = 0.56
c) P(X + Y = 0)=0.3
d) P(X + Y <= 1)=0.53
Step-by-step explanation:
We have to construct the joint probability table with the marginal probabilities of X and Y.
X can take values from 0 to 2, and Y can take values from 0 to 4.
We can calculate each point of the joint probability as:

Then, the joint probabilities are:
X=0	Y=0	Px=0.5	Px=0.6	P(0,0)=0.3
X=0	Y=1	Px=0.5	Px=0.1	P(0,1)=0.05
X=0	Y=2	Px=0.5	Px=0.05	P(0,2)=0.025
X=0	Y=3	Px=0.5	Px=0.05	P(0,3)=0.025
X=0	Y=4	Px=0.5	Px=0.2	P(0,4)=0.1
X=1	Y=0	Px=0.3	Px=0.6	P(1,0)=0.18
X=1	Y=1	Px=0.3	Px=0.1	P(1,1)=0.03
X=1	Y=2	Px=0.3	Px=0.05	P(1,2)=0.015
X=1	Y=3	Px=0.3	Px=0.05	P(1,3)=0.015
X=1	Y=4	Px=0.3	Px=0.2	P(1,4)=0.06
X=2	Y=0	Px=0.2	Px=0.6	P(2,0)=0.12
X=2	Y=1	Px=0.2	Px=0.1	P(2,1)=0.02
X=2	Y=2	Px=0.2	Px=0.05	P(2,2)=0.01
X=2	Y=3	Px=0.2	Px=0.05	P(2,3)=0.01
X=2	Y=4	Px=0.2	Px=0.2	P(2,4)=0.04
We can write it in the form of a matrix:

b) From the joint probability P(X<= 1 and Y <= 1) is equal to

We can calculate P(X<= 1) * P(Y<=1)

Both calculations give the same result.
c) Probability of no violations

d) P(X + Y <= 1)
