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  1. MHB Apprentice
    lucytranxx's Avatar
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    #1
    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 dont understand what X is suppose to be???

  2. MHB Craftsman

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    #2
    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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