Finding the PMF of Y with Known X

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Discussion Overview

The discussion revolves around finding the probability mass function (PMF) of a random variable \(Y\) defined as a transformation of another random variable \(X\). The transformation is given by \(Y = \sin\Big(\frac{\pi}{2X}\Big)\), with a specified PMF for \(X\) where \(p_X[k] = \frac{1}{5}\) for \(k = 0,1,\ldots, 4\). The scope includes theoretical aspects of probability distributions and transformations of random variables.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about methods to find the PMF of \(Y\) given the transformation from \(X\).
  • Another participant suggests that there may be multiple approaches to solving the problem and references theorems related to characteristic functions of transformed random variables.
  • Some participants express confusion regarding the interpretation of the PMF \(p_X[k]\) and whether it implies \(P\{X=k\} = \frac{1}{5}\) for \(k = 0,1,2,3,4\).
  • Concerns are raised about the definition of \(Y\) when \(X=0\), indicating potential difficulties in the transformation.

Areas of Agreement / Disagreement

Participants express uncertainty about the problem's resolution, with some indicating that the problem appears to be solved while others note the absence of a clear solution. There is no consensus on the approach to finding the PMF of \(Y\).

Contextual Notes

Participants highlight the challenge of defining \(Y\) when \(X=0\) and the implications this has for the transformation. There is also ambiguity regarding the interpretation of the PMF for \(X\).

Dustinsfl
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How do I find the PMF of Y when I know X?
\[
Y = \sin\Big(\frac{\pi}{2X}\Big)
\]
and
\[
p_X[k] = \frac{1}{5}
\]
for \(k = 0,1,\ldots, 4\).
 
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Maybe there're several ways to solve this problem. It's clear that $Y$ is a transformation of $X$. So have you seen any theorems about characteristic funtions of transformated random variables?

EDIT:
I now see that this problem is already solved.
 
Siron said:
...EDIT:
I now see that this problem is already solved.

Well, the [SOLVED] prefix has been added, but I see no solution...
 
dwsmith said:
How do I find the PMF of Y when I know X?
\[
Y = \sin\Big(\frac{\pi}{2X}\Big)
\]
and
\[
p_X[k] = \frac{1}{5}
\]
for \(k = 0,1,\ldots, 4\).

There is a point not clear: writing $\displaystyle p_X[k] = \frac{1}{5}, k= 0,1,2,3,4$ means writing $\displaystyle P\{X=k \} = \frac{1}{5}, k= 0,1,2,3,4$?...

Kind regards

$\chi$ $\sigma$
 
chisigma said:
There is a point not clear: writing $\displaystyle p_X[k] = \frac{1}{5}, k= 0,1,2,3,4$ means writing $\displaystyle P\{X=k \} = \frac{1}{5}, k= 0,1,2,3,4$?...

Kind regards

$\chi$ $\sigma$

The reason of my question is that for X=0 defining $\displaystyle Y = \sin \frac{\pi}{2\ X}$ is a little difficult task (Dull)...

Kind regards

$\chi$ $\sigma$
 

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