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I was practicing for my next stochastic exam and I was doing some exercises.

I have noticed that I still find it difficult to model a mathematical model from a problem.

Let's take the following problem for example:

From an urn with $n$ balls numbered from $1$ to $n$, $k ≥ 1$ balls are drawn random and and then put back inside.

$X$ be the largest drawn number among the $k$ drawn.

Let's assume I have to model a probability space.

Than for the P-Space $(\Omega ,F,P)$ I would write:

$\Omega=\left \{ (w_1,...,w_k)| w_i \in\left \{ 1,...,n \right \},i \in \left \{ 1,...,k \right \} \right \} $ $\\F=Pot(\Omega) \\\text{and } X=\left \{ w_i \in(w_1,...,w_k)|w_i \geq w_j, \text{for all } j \neq i \right \}$

Now how can I find the probability measure $P$ if I don't know what I'm going to do with this model.

I don't know if it is clear what I mean: how can I find a probability measure if I don't know which event I have to find?

MarcoDJ01
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    "..$k\geq 1$ balls are drawn and then put back inside.." Does this happen after drawing all of them? Or is each drawn ball put back immediately? I suspect the second option but would like more certainty. – drhab Sep 27 '22 at 09:36
  • Each drawn ball is put back immediately – MarcoDJ01 Sep 27 '22 at 09:40
  • Yes, formally you must be capable of making models but I tend to say: do not worry about it too much. To make clear what I mean see this answer. – drhab Sep 27 '22 at 10:07

1 Answers1

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Modeling.

If $[n]=\{1,\dots,n\}$ then $\Omega=[n]^k$ does fine with equiprobable outcomes $\omega=(\omega_1,\dots,\omega_k)$.

There are $n^k$ outcomes so that on base of:$$1=P(\Omega)=\sum_{\omega\in\Omega}P(\{\omega\})$$we find $P(\{\omega\})=n^{-k}$.

That gives birth to the complete probability measure as a function on $\mathcal F=\mathcal P(\Omega)$ : $$P(A)=\sum_{\omega\in A}P(\{\omega\})=n^{-k}|A|$$

After modeling.

For $i=1,\dots,k$ define $X_i(\omega):=\omega_i$ and $X=\max\{X_1,\dots,X_k\}$.

The $X_i$ are iid so that:$$P(X\leq m)=P(X_1\leq m,\dots,X_k\leq m)=P(X_1\leq m)^k$$

Can you find $P(X_1\leq m)$ yourself?

drhab
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