MHB Transformation of random variable

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To express the probability function of the transformed random variable Y=g(X), where X is a discrete random variable with equal probabilities, the probability mass function can be derived as f_Y(y)=|g^{-1}(y)|/n. This formula indicates that the probability of Y taking a specific value y depends on the number of pre-images of y under the function g, denoted as |g^{-1}(y)|. The discussion highlights that without additional information about the function g, further simplification of the probability function is not possible. Clarification is needed regarding the notation used, as it may lead to confusion in understanding the relationship between Y and its corresponding probabilities. Understanding the transformation of random variables is crucial for accurate probability calculations.
WMDhamnekar
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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?

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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 $$f_Y(y)=|g^{-1}(y)|/n$$. I don't think this can be simplified unless we know more about $g$.
 

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