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Suppose, it is stated that $X$ is a random variable which has a symmetric distribution.

Now, let us consider $Y$ to be a random variable which has a distribution given by : $Y = g(X)$, where '$g$' is a non-linear function.

By a non-linear function, I mean that it can be either a logarithmic or a cubic function, i.e., one can consider either $Y = log(X)$ or $Y = X^3$.

If this is so, then what can we say about the skewness of the distribution of Y ? Is it symmetric or positively skewed or negatively skewed ?

Dwaipayan Gupta
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    You cannot say anything in general if your understanding of "nonlinear" is the conventional one (meaning that it cannot be expressed as $Y=\alpha+\beta X$ for constants $\alpha$ and $\beta$): $Y$ could be not skewed or skewed in either direction. Do you really mean that "nonlinear" is solely the logarithm or the third power? – whuber Jan 28 '17 at 19:11
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    In the case of taking the logarithm of a positive random variable, see the discussion here (for example - there are other somewhat relevant discussions). For the cube it depends on where the variable "lives"; it might go either one way or the other way, or it might remain symmetric, or it might do something more complicated (become asymmetric without being clearly skew in either direction) – Glen_b Jan 29 '17 at 02:44
  • Yes, @whuber by "non-linear", I mean solely the logarithmic or the cubic function.... More specifically, the cubic function... – Dwaipayan Gupta Jan 29 '17 at 04:51
  • @Glen_b you said "For the cube it depends on where the variable "lives"". Could you explain to me as to what you meant by "lives" ? – Dwaipayan Gupta Jan 29 '17 at 04:51
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    It depends on whether the variate is positive, negative, symmetric about zero or something else. For a symmetric case, the location of the center of symmetry is important (if it were asymmetric, the story is more complex and depends on the distribtution shape). If the link I gave for the log is helpful, a similar set of analyses covers the cubic case. – Glen_b Jan 29 '17 at 06:13

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