| layout |
mathjax |
author |
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date |
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theorem |
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proof_id |
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username |
proof |
true |
Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2021-08-29 22:13:00 -0700 |
Probability mass function of an invertible function of a random vector |
General Theorems |
Probability theory |
Probability mass function |
Probability mass function of invertible function |
| authors |
year |
title |
in |
pages |
url |
Taboga, Marco |
2017 |
Functions of random vectors and their distribution |
Lectures on probability and mathematical statistics |
retrieved on 2021-08-30 |
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P253 |
pmf-invfct |
JoramSoch |
Theorem: Let $X$ be an $n \times 1$ random vector of discrete random variables with possible outcomes $\mathcal{X}$ and let $g: ; \mathbb{R}^n \rightarrow \mathbb{R}^n$ be an invertible function on the support of $X$. Then, the probability mass function of $Y = g(X)$ is given by
$$ \label{eq:pmf-invfct}
f_Y(y) = \left{
\begin{array}{rl}
f_X(g^{-1}(y)) ; , & \text{if} ; y \in \mathcal{Y} \\
0 ; , & \text{if} ; y \notin \mathcal{Y}
\end{array}
\right.
$$
where $g^{-1}(y)$ is the inverse function of $g(x)$ and $\mathcal{Y}$ is the set of possible outcomes of $Y$:
$$ \label{eq:Y-range}
\mathcal{Y} = \left\lbrace y = g(x): x \in \mathcal{X} \right\rbrace ; .
$$
Proof: Because an invertible function is a one-to-one mapping, the probability mass function of $Y$ can be derived as follows:
$$ \label{eq:pmf-invfct-qed}
\begin{split}
f_Y(y) &= \mathrm{Pr}(Y = y) \\
&= \mathrm{Pr}(g(X) = y) \\
&= \mathrm{Pr}(X = g^{-1}(y)) \\
&= f_X(g^{-1}(y)) ; .
\end{split}
$$