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## Homework Statement

Let's say one measures an observation with absolute precision using a set of indicator functions, in a sample space with 3 elements. Then these indicator functions will be the usual discrete indicator functions of the form

[tex] I_n(j) = \begin{cases} 1 & \text{ if } n = j\\ 0 & \text{ otherwise } \end{cases} [/tex]

Now let's say that your measurement method is flawed, such that 50% of the time your observation is accurate to the event, and the other 50% of the time your observation is random, though uniformly distributed over all three elements in the sample space. What will the membership function be in this case.

## Homework Equations

The membership function is just a generalized indicator function over "fuzzy" sets. That is, for every member in the sample space, they take on a value between 0 and 1 (though still sum up to 1 over all space).

## The Attempt at a Solution

It seems to me we want to say something like [itex] I_n(j) [/itex] will be the `probability' that we observe event n, given that event j happened. This would amount to:

Probability of observing n given j happened + probability of observing n given that j didn't happen.

I boil this down to

P(accurate that j occured)*P(measured n) + P(inaccurate)*P(measured n) + P(accurate)*P(did not measure n) + P(inaccurate in observing j)*P(did not measure n)

But I'm really not sure about this