Let X be the random variable that records the number of “heads” when two coins are tossed. Let Y be a random variable with the discrete uniform distribution on the probability space {1, 2, 3}. Assume that X and Y are independent. Let U be the random variable defined by U = X + Y . Find the probability distribution for the random variable U.
How is the probability distribution calculated in this case?
First of all check the support of the new rv $Z=X+Y$
$z \in \{1;2;3;4;5\}$
to calculate the probability of each $z$ the easiest way is to construct a $2 \times 2$ table with inside the table the probability of the joint distribution (easy because of independence) then detect all the possible cases for each z
$$\mathbb{P}[Z=2]=\mathbb{P}[X=0;Y=2]+\mathbb{P}[X=1;Y=1]=\frac{1}{4}\times\frac{1}{3}+\frac{2}{4}\times\frac{1}{3}=\frac{3}{12}$$
now you have only to calculate all the other probabilities of $z \in Z$