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Transformation of random variable

Dhamnekar Winod

Active member
Nov 17, 2018
100
Hello,

A discrete random variable X takes values $x_1,...,x_n$ each with probability $\frac1n$. Let Y=g(X) where g is an arbitrary real-valued function.


I want to express the probability function of Y(pY(y)=P{Y=y}) in terms of g and the $x_i$
How can I answer this question?

If any member knows the correct answer, he/she may reply with correct answer.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,488
The notation Y(pY(y)=P{Y=y}) is confusing. For one, $Y$ accepts as argument elements of $\{x_1,\ldots.x_n\}$ and not equalities. if you need the probability mass function of $Y$, it is \(\displaystyle f_Y(y)=|g^{-1}(y)|/n\). I don't think this can be simplified unless we know more about $g$.