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Thread: Deriving Probability Function from Non-Independent SubFunctions

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    Deriving Probability Function from Non-Independent SubFunctions

    Howdy All,

    I attached a picture (inserted at bottom of post), to help explain what I'm trying to accomplish. In general, I don't need "the answer" so much as I need guidance on where to look to help me figure out the answer (concepts to google, book sections, PDF tutorials, whatever).

    The problem I have is the following:
    * I have 3 lines defined in a common spatial coordinate frame, (X,Y). The lines are L_A, L_B, and L_C (shown in black in the picture).
    * I have three probability distribution functions (F) associated with those lines, such that the probability (Z) associated with a given point, P, on a given line, L, can be found via Z = F(P_x,y)
    ** For a given point on either line, this yields the following relationships:
    *** Z_Ax,Ay = F_A(P_Ax,Ay) ... for a point on line L_A (green lines)
    *** Z_Bx,By = F_B(P_Bx,By) ... for a point on line L_B (red lines)
    *** Z_Cx,Cy = F_C(P_Cx,Cy) ... for a point on line L_C (blue lines)
    * For each line, L, I have an overall probability associated with it that is NOT independent of the other lines. For instance, line L_A might have an overall probability of 2%, while line L_C might have a probability of 9%, with line L_B having a probability of 42%.
    ** This implies that there is a continuous probability function associated with all lines sweeping from L_A through to L_C (and continuing off into tails). So, there could be a line between L_A and L_B, in the example above, that has a probability of 33%, which would imply that the new line lies closer to L_B rather than L_A.
    What I am trying to do is derive a combined probability distribution function, F_T, which would give a Z_T value for any given point X,Y coordinate within the space shown. The combined function would need to account for the fact that the points input into, F_A, F_B, F_C, etc. have a probability associated with them as well (the "line probabilities" described above for L_A, L_B, and L_C.

    So, in my mind's eye, I'm imagining the individual probability functions, F_A, F_B, and F_C as looking like 3 "peaks" or "mountains," the height of which is scaled by Z_A, Z_B, and Z_C respectively, if viewed from the perspective of someone standing on the page, looking down one of the lines. So in the example above, the Z_B mountain would be higher than the Z_C mountain, both of which would be higher than the Z_A mountain.

    What I need to do is scale those peaks by their associated "line probabilities," and then use those scaled peaks to fit a single "encompassing" mountain, the contour of which would be defined by the "scaled peaks" of the three sub-probabilities. This "encompassing mountain" then, would be a final probability distribution function (F_T) which I could sample for any given X,Y coordinate, that would give me the total probability, Z_T, associated with that X,Y coordinate.

    I should note, this isn't a homework problem. This is actually something I'm trying to solve for my job. I just don't have the depth in statistics to recall how to properly "blend" probability functions like this.

    Any help would be very much appreciated, even if it is just a list of topics I should read about on Wolfram Alpha to answer this question for myself.


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