3
$\begingroup$

For continuous distribution (on R) the probability of a single point is $0$. So I'm not sure what does it mean to sample $M$ elements from a continuous distribution.

Let say there is a continuous distribution D and there is a number z and a function f such that: $f(x)=1$ for $x < z$ except of a finite number of cases.

$f(x)=0$ for $x \ge z$ except of a finite number of cases.

$D(x < z) > 0 , D(x \ge z) > 0$

So if I have a random sample $(X_1, \dots ,X_m)$ from $D$, And assume $X_1>z$ , can I conclude that $f(X_1)=0$ ?
And assume $f(X_2)=1$ , can I conclude that $X_2 < z$ ?

$\endgroup$
1
$\begingroup$

Sampling $m$ elements simply means obtaining $m$ values from $X_i$, call them $x_i$, according to the distribution $D$, exactly as for the discrete case. If you prefer, think about it as choosing a number which has the correct probability of being in any interval.

If your observation $x_i$ of $X_i$ satisfies $x_i>z$, then with probability one it satisfies the result you give. Strictly the statement is $P(f(X_i)=0|X_i>z) =1$. That is, with probability 1 your conclusion holds. Similarly for $x_2$.

The key points are

  • The fact that something happens with probability 0/1 doesn't mean it's impossible/certain.

  • You make statements about specific observed values of random variables by referring to particular observations, and statements about probabilistic deductions you can make given random variables by using conditional probabilities.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.