I generated random numbers in Excel $(0 < y < 2)$ such that their distribution in a histogram has the shape of a right triangle and is skewed right, not left. I want a transformation or series of transformations to keep the same limits ($0$ and $2$) but change the shape to a triangle having zero skewness. Also, there must be one-to-one mapping. For example, $y = 1$ before transformation is $y = 1$ after transformation. Is this even possible? I can get close with $[(y^a - 1)/a] + b$, where $a = 0.875$ and $b = 1.1$, but might be going down the wrong path. I'm not a mathematician obviously, so any ideas are appreciated.
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Sorry, I made mistakes in my original post. I am revising it here and adding some images.
I generated random numbers in Excel (0 <= y <= 1) such that their distribution in a histogram has the shape of a right triangle and is skewed right.
I want a transformation or series of transformations to change the limits (0 and 2 instead of 0 and 1) and the histogram shape (zero skewness). The distribution needs to be symmetric about y = 1 but not necessarily Gaussian.
Also, there must be linear mapping. For example, y = 0.5 and y = 0.75 before transformation are, respectively, y = 1 and y = 1.5 after transformation.
I don't think that all this is possible. Am I wrong?
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Keeping the limits (0 and 1) or changing to any other ones, I don't think that it's possible to satisfy both remaining requirements (symmetry about the limits' midpoint; linear mapping). For example, (y^a)/a, where a = 0.575, satisfies zero skewness but not linear mapping.
