MHB Bernoulli Trials and Probability

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The discussion focuses on the computation of the joint probability distribution of a random variable X, defined as the sum of 5 independent Bernoulli trials with a success probability r, which can take on three specified values. The joint distribution of X and R is derived using the law of total probability, acknowledging that X follows a binomial distribution given R. The marginal distribution function of X, along with its unconditional mean and variance, can be computed from the joint distribution. Clarification is sought regarding the variable Y, as it is not defined in the context of the problem. The conversation emphasizes the relationship between the random variables and the need for clear definitions in probability discussions.
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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?
 
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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.
 
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