MHB Bernoulli Trials and Probability

[lucytranxx]

Let X be a random variable defined as the sum of 5 independent Bernoulli trials in which the probability of each Bernoulli taking the value 1 is given by r. Suppose that prior to the 5 Bernoulli trials, r is chosen to take one of three possible values with the following probabilities:
R=r P(R=r)
0.1 0.2
0.5 0.5
0.4 0.3

(a) Compute the joint probability distribution of X and R Are Y and R independent? Provide your reasoning.


(b) Compute the marginal distribution function of X and the unconditional mean and variance of Y

this was a question in one of the textbooks but i don't understand what X is suppose to be?

 

[Siron]

In general, the sum of $n$ independent Bernoulli trials where the succes probability is $r$, follows a binomial distribution with parameters $n$ and $r$. Hence, $X \sim \mbox{Binomial}(5,r)$. Further, note that $X$ and $R$ are both discrete random variables. The (marginal) distribution of $R$ is given, however the distribution of $X$ depends on $R$. To find the distribution of $X$, you can make use of the law of total probability, that is,
$$\mathbb{P}(X=x) = \sum_{r} \mathbb{P}(X = x \ | \ R = r) \mathbb{P}(R = r)$$
where $x \in \{0,\ldots,n\}$.

Question: what is $Y$? I do not see any description.