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I have a probability question: How can I represent the probability of an event occurring at a specific {time,place}? How are the time and place represented?

Example: I want to represent the occurrence of neighborhood dogs peeing in my neighborhood

Let's say I can count the number of time of dogs peeing, and the state space is of {time,place}, let's say per week and per area of neighborhood.

I want to narrow the event to a specific occurrence: {time,place: time $=$ only on Sunday; place $=$ only in my backyard}

If the probability of the dogs peeing event at a specific place is defined as:

  • $Ap$ - the event of dog peeing at a specific place
  • $P(Ap=x)$ - the probability of dog pee at a specific place $x$

and the probability of dog peeing event on a specific time:

  • $At$ - the event of dog peeing on a specific time $t$
  • $P(At=t)$ - the probability of dog pee on a specific time $t$

How can I represent the probability of a pee occurring at a specific {time,place}?

P.S.

How to treat the time, place ?

Are time,place considered joint random variables $Xt,Xp ? P(Xt=t,Xp=x)$

If so are they independent? $P(Xt=t,Xp=p)$ = $P(Xt=t)*P(Xp=x)$

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One way you can represent the probability by adding 'not peeing' as a place.

So you can write the stochastic process as $X = \{ X_t: t \in T \}$ with the discrete states $x_1,x_2,\ldots,x_p$ and 'not peeing'.

The representation of the dog pee at a specific place x on time t is $P[X_t=x]$

The probability of dog pee on a specific time $t$ can be written as

$P[X_t \ne \text{'not peeing'}] = 1 - P[X_t = \text{'not peeing'}]$.

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