What mean the notation $p_{t,s}(x,dy)$?
1
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In my course (see picture below) (under the formula (0.1.3)), they say that $p_{s,t}(x,dy)$ is a regular version of the conditional probability distribution of $X_t$ given $X_s$. Could someone tell me what does it mean ? Does is mean that $$mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy) ?$$
probability measure-theory
asked Jan 2 at 14:02
NewMathNewMath
4059
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add a comment |
1
$begingroup$
In my course (see picture below) (under the formula (0.1.3)), they say that $p_{s,t}(x,dy)$ is a regular version of the conditional probability distribution of $X_t$ given $X_s$. Could someone tell me what does it mean ? Does is mean that $$mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy) ?$$
probability measure-theory
asked Jan 2 at 14:02
NewMathNewMath
4059
$endgroup$
$begingroup$
In this particular situation there isn't really any reason to write it like that rather than as $p_{s,t}(x,A)$, which is a probability measure characterizing the probability of transitioning from a fixed point $x$ to a set $A$ as time goes from $s$ to $t$. In other situations this sort of notation can help with defining more complicated measures.
$endgroup$
– Ian
Jan 2 at 14:18
$begingroup$
@Ian: Thank you. So my interpretation is correct ? i.e. $p_{s,t}(x,A)=mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy)$ ?
$endgroup$
– NewMath
Jan 2 at 14:20
$begingroup$
That's correct.
$endgroup$
– Ian
Jan 2 at 14:22
add a comment |
1
1
1
$begingroup$
In my course (see picture below) (under the formula (0.1.3)), they say that $p_{s,t}(x,dy)$ is a regular version of the conditional probability distribution of $X_t$ given $X_s$. Could someone tell me what does it mean ? Does is mean that $$mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy) ?$$
probability measure-theory
asked Jan 2 at 14:02
NewMathNewMath
4059
$endgroup$
In my course (see picture below) (under the formula (0.1.3)), they say that $p_{s,t}(x,dy)$ is a regular version of the conditional probability distribution of $X_t$ given $X_s$. Could someone tell me what does it mean ? Does is mean that $$mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy) ?$$
probability measure-theory
probability measure-theory
asked Jan 2 at 14:02
NewMathNewMath
4059
asked Jan 2 at 14:02
NewMathNewMath
4059
asked Jan 2 at 14:02
NewMathNewMath
4059
asked Jan 2 at 14:02
NewMathNewMath
4059
asked Jan 2 at 14:02
NewMathNewMath
4059
4059
$begingroup$
In this particular situation there isn't really any reason to write it like that rather than as $p_{s,t}(x,A)$, which is a probability measure characterizing the probability of transitioning from a fixed point $x$ to a set $A$ as time goes from $s$ to $t$. In other situations this sort of notation can help with defining more complicated measures.
$endgroup$
– Ian
Jan 2 at 14:18
$begingroup$
@Ian: Thank you. So my interpretation is correct ? i.e. $p_{s,t}(x,A)=mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy)$ ?
$endgroup$
– NewMath
Jan 2 at 14:20
$begingroup$
That's correct.
$endgroup$
– Ian
Jan 2 at 14:22
add a comment |
$begingroup$
In this particular situation there isn't really any reason to write it like that rather than as $p_{s,t}(x,A)$, which is a probability measure characterizing the probability of transitioning from a fixed point $x$ to a set $A$ as time goes from $s$ to $t$. In other situations this sort of notation can help with defining more complicated measures.
$endgroup$
– Ian
Jan 2 at 14:18
$begingroup$
@Ian: Thank you. So my interpretation is correct ? i.e. $p_{s,t}(x,A)=mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy)$ ?
$endgroup$
– NewMath
Jan 2 at 14:20
$begingroup$
That's correct.
$endgroup$
– Ian
Jan 2 at 14:22
$begingroup$
In this particular situation there isn't really any reason to write it like that rather than as $p_{s,t}(x,A)$, which is a probability measure characterizing the probability of transitioning from a fixed point $x$ to a set $A$ as time goes from $s$ to $t$. In other situations this sort of notation can help with defining more complicated measures.
$endgroup$
– Ian
Jan 2 at 14:18
$begingroup$
In this particular situation there isn't really any reason to write it like that rather than as $p_{s,t}(x,A)$, which is a probability measure characterizing the probability of transitioning from a fixed point $x$ to a set $A$ as time goes from $s$ to $t$. In other situations this sort of notation can help with defining more complicated measures.
$endgroup$
– Ian
Jan 2 at 14:18
$begingroup$
@Ian: Thank you. So my interpretation is correct ? i.e. $p_{s,t}(x,A)=mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy)$ ?
$endgroup$
– NewMath
Jan 2 at 14:20
$begingroup$
@Ian: Thank you. So my interpretation is correct ? i.e. $p_{s,t}(x,A)=mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy)$ ?
$endgroup$
– NewMath
Jan 2 at 14:20
$begingroup$
That's correct.
$endgroup$
– Ian
Jan 2 at 14:22
$begingroup$
That's correct.
$endgroup$
– Ian
Jan 2 at 14:22
add a comment |
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$begingroup$
In this particular situation there isn't really any reason to write it like that rather than as $p_{s,t}(x,A)$, which is a probability measure characterizing the probability of transitioning from a fixed point $x$ to a set $A$ as time goes from $s$ to $t$. In other situations this sort of notation can help with defining more complicated measures.
$endgroup$
– Ian
Jan 2 at 14:18
$begingroup$
@Ian: Thank you. So my interpretation is correct ? i.e. $p_{s,t}(x,A)=mathbb P{X_tin Amid X_s=x}=int_A p_{s,t}(x,dy)$ ?
$endgroup$
– NewMath
Jan 2 at 14:20
$begingroup$
That's correct.
$endgroup$
– Ian
Jan 2 at 14:22