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I've recently get to the notion of very basic of statistics.

The statistics is trying to assign a "Random Values(mostly in $\Bbb R$)to the given specific event.

So the defining of random variable f would be :

$f: \frak C \rightarrow \Bbb R$ where $f(c) = x $ $\;(c \in \frak {C}$ and $x \in \Bbb R)$ and $f $ is injective.

I think my book does not explicitly denoted that f must be injective however, I think if we omit the injectivity of the RV fucntion, there would be no chance of assigning "the event" exclusively to one "exclusive number"

What do you think of this injectivity? Is my understanding correct? If not, I need your advice to fix it.

Daschin
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1 Answers1

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It does not have to be injective and it is also practical in many cases.

Consider your $\frak C$ to be all children to be born and each $c\in \frak C$ is a child. We can define several variables such as sex, let's code $f(c)=0$ for boys and $f(c)=1$ for girls.

Each child is unique (each parent will confirm that), but the introduced random variable has only two values.

Intuitively speaking, statistical reasoning even tries to be in most cases "anti-injective" because it wants to transform unique objects to a common language and draw common conclusions such as the probability of being a boy is less than 50%. This is valid from the scientific perspective, but sometimes this may lead to ethical dilemmas if someone proclaims 99% of people want to kill the 1% minority and considers it as an argument for doing so. In that case, we have to simply come back to $\frak C$ and consider the uniqueness of each human being rather from ethical or philosophical than purely statistical perspective.

Karel Macek
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