The definition of $X|(Y=y)$

conditional probabilitydefinitionprobabilityprobability theory

Suppose $S$ is a sample space (the set of all outcomes $\omega_i$) for an experiment. A random variable $X$ is defined as a real-valued function which maps elements from the sample space to real numbers, i.e. $X:S\to \mathbb R$.

Discrete Random variable:

The definition of the conditional probability mass function of $X$ given $Y=y$ is $$\mathbb P(X=x|Y=y)=\frac{\mathbb P(X=x, Y=y)}{\mathbb{P}(Y=y)} .$$

Question: In lecture slides I have seen the notation, for example, that $X|(Y=y) \sim \text{Bin}(m, \lambda).$
What is the definition of $X|(Y=y)$? Is it a random variable itself with a restricted sample space? Maybe $X|(Y=y): \{\omega\in S: Y(\omega)=y \} \to \mathbb R$?

What would be the definition of $X|(Y=y)$ for $X$ and $Y$ being continuous random variables?

(Note: If it isn't a random variable, then how can we talk about it's distribution and expected value?)

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Summarising the very helpful comments from @Nap D. Lover and @d.k.o. - In the original theory of conditional probability, there is no such definition of a "conditional random variable."

Before addressing the notation, a thought about the "requirement" of a conditional random variable

The purpose of a conditional distribution, $\mathbb P(X=x|Y=y)$, is a way to "recalibrate" the probability assignment/distribution for $X$, given we received information about $Y$. (Which intuitively, could be the probability distribution of the temperature $X$ as $\mathbb P(X=x)$ vs. the probability distribution of the temperature $X$, given the humidity $Y$ was $y$, being $\mathbb P(X=x|Y=y)$). It is still a probability distribution designed for the random variable $X$, just "recalibrated" to better model the "true" probabilities for the given situation.
So I guess, in a way, a new random variable for a "conditional random variable" is not really necessary. While it is possible to define a random variable $X_y$ living on a new restricted sample space, maybe it moves away from the idea of this distribution being "rediagnosis" of what the probability distribution of $X$ should be, given the new "symptoms" ($Y=y$).
Hence it makes sense to only need Conditional distributions and Conditional expectation (The expected value of $X$, but weighted in a different way to account for the new information) etc, and not a new random variable itself.

The notation: So the interpretation of the notation can be left as what @d.k.o. said in the very first comment, $X|(Y=y) \sim \text{Bin}(m, \lambda)$ is just shorthand notation for saying "The distribution of $X$, conditioned on $Y=y$, is (from the definition in the question) $\text{Bin}(m, \lambda)$.

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Probability Theory – Precise Definition of the Support of a Random Variable

I am not entirely convinced with the line the sample space is also called the support of a random variable

That looks quite wrong to me.

What is even more confusing is, when we talk about support, do we mean that of $X$ or that of the distribution function $Pr$?

In rather informal terms, the "support" of a random variable $X$ is defined as the support (in the function sense) of the density function $f_X(x)$.

I say, in rather informal terms, because the density function is a quite intuitive and practical concept for dealing with probabilities, but no so much when speaking of probability in general and formal terms. For one thing, it's not a proper function for "discrete distributions" (again, a practical but loose concept).

In more formal/strict terms, the comment of Stefan fits the bill.

Do we interpret the support to be

- the set of outcomes in Ω which have a non-zero probability,
- the set of values that X can take with non-zero probability?

Neither, actually. Consider a random variable that has a uniform density in $[0,1]$, with $\Omega = \mathbb{R}$. Then the support is the full interval $[0,1]$ - which is a subset of $\Omega$. But, then, of course, say $x=1/2$ belongs to the support. But the probability that $X$ takes this value is zero.

Conditional Variance: Is it a random variable

By definition, $$ \textrm{Var}(Y\mid X):=\mathbb E\bigl[\bigl(Y-\mathbb E(Y\mid X)\bigr)^2\mid X\Bigr]=\mathbb E(Y^2\mid X)-\mathbb E(Y\mid X)^2. $$ Thus, the conditional variance is a random variable, in the same way that the conditional expectation $\mathbb E(Y\mid X)$ is. Conceptually, the variance is the "same type of object" as the expectation, in this regard.

Now, one may also consider an event $A\subseteq \Omega$ (the sample space) and ask what is $\textrm{Var}(Y\mid A)$. And it follows the exact same behavior as the conditional expectation, namely that we define $$ \textrm{Var}(Y\mid A):=\mathbb E\bigl[\bigl(Y-\mathbb E(Y\mid A)\bigr)^2\mid A\Bigr]=\mathbb E(Y^2\mid A)-\mathbb E(Y\mid A)^2. $$

By definition, $$\mathbb E(Y\mid A):=\frac{\mathbb E(Y\cdot 1_A)}{\mathbb E(1_A)},$$ where $1_A$ denotes the indicator of the set $A$. It is a random variable taking the value $1$ on $A$ and $0$ off $A$. Note also that $\mathbb E(1_A)=\mathbb P(A)$, I just wrote it that way in the denominator of the formula for consistency with the numerator.

Per the discussion below, there was an even more basic question that I should clarify. A random variable is a function from the sample space $\Omega$ to the real numbers. This means it assigns a real number to each element $\omega\in \Omega$. On the other hand, when we condition on an event we obtain a set function on $\Omega$, or in other words, a function that assigns values to subsets of $\Omega$ and not to individual elements of $\Omega$. In this case, being even more precise, we have a partially defined set function which means that not every subset is assigned a value - it is only those subsets which are measurable and are assigned a positive measure for which the conditional variance is defined.

To compare and contrast the two types of mathematical objects, conditional variance with respect to a random variable is a function from $\Omega$ to $\mathbb R$, whereas conditional variance with respect to an event is a partially defined function from $P(\Omega)$ to $\mathbb R$ (the power set of $\Omega)$.

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Probability Theory – Precise Definition of the Support of a Random Variable

Conditional Variance: Is it a random variable

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