is it possibile to construct a random variable such that the avarage of its values matching 0s in a binary sequence is double than its values matching 1s? For instance given a binary random sequence of 0s and 1s assign to each value of the sequence a value such that the avarage matching the 0s is double of the avarage matching the 1s.
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I'm not sure what you really want, but consider this hierarchy model:
We have a random variable $X$ with finite mean $E[X] = \mu$, and we can have a random variable called $Y$ (which can be independent to $X$ if you like) such that $E[Y] = 2\mu$. We have another Bernoulli random variable (a coin flip sequence) called $B$.
Now we define yet another random variable $W$ such that:
$$W = \begin{cases} X & \text{when $B = 0$, with probability $p$} \\ Y & \text{when $B = 1$, with probability $1-p$} \end{cases}$$
Then the observations (samples) of $W$ is the desired sequence.
For example, $X$ can have a Normal distrbiution $\mathcal{N}(\mu=3, \sigma^2 = 1)$ and $Y$ can follow a Gamma distribution with shape parameter $6$ and scale parameter $1$ (such that its mean is $6$).
Lee David Chung Lin
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The problem is the assignment of X and Y before the value of B is Known. For instance is possible to assign X or Y when the value of the next B is not yet Known.
francesco
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Sorry I didn't see your "response" here. Next time please comment under my post to let me know. Anyway, this clarification should go into your question post instead of being here. Please understand that your question is interesting, but it has been overlooked mainly because you didn't bother to type up the question with enough context and clarity. – Lee David Chung Lin Apr 25 '18 at 12:03
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You can click the tiny "edit" near the bottom of your question to update the content. – Lee David Chung Lin Apr 25 '18 at 12:04
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At the same time, you seem to misunderstand how a hierarchy model works. In my answer post, all the four things $B$, $X$, $Y$, and $W$ are random variables. Nothing is ever known "beforehand", and $W$ is well-defined with the desired property. Having said that, your description is not very clear so even now I'm still really just guessing what you want. – Lee David Chung Lin Apr 25 '18 at 12:08
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My question informally is this. I have a coin to flip. Simply i must choice a number before flipping. After that number is associated to the result (tail/head). – francesco Apr 26 '18 at 14:26
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Then please add this description in the question post (so that it is more comprehensible and self-contained). Again, you can click the tiny "edit" near the bottom of your question to update the content. This is not just for me but for other people who might see this question. You cannot expect people to go through the conversation between you and I in the comments. – Lee David Chung Lin Apr 26 '18 at 14:35