Generalized stochastic equation
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We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let $X_n$ be the number of individuals that have received the message at time $n$, then assume that the $k^th$ individual with the message at time $n$, passes it on to $A_k,n$ other individuals. We then have the following recursion for $X_n$:
$$X_n=sum_k=1^X_n-1A_k,n$$
Further assume that the random variables $A_k,n$ are independent and identically distributed for all $k$ and $n$. Use the notation $a$ and $v$ for the (common) mean and variance $A_k,n$. We assume $a > 1$ in all questions.
Calculate $E[X_n|X_n−1]$ and $E[X_2|X_n−1]$.
Calculate $E[X_n|X_0]$ and $E[X_2|X_0]$.
(tip: calculate first $E[X_2|X_n−2]$,
$E[X_2|X_n−3]dots$ before you try to get the
expression for $E[X_2|X_0]$)
Can anyone help me how to solve it when i dont have value for $X_0$ and $k$ starts from $1$?
stochastic-processes
edited Aug 7 at 21:34
mrtaurho
698219
asked Aug 7 at 18:43
New comer
6
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up vote
0
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We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let $X_n$ be the number of individuals that have received the message at time $n$, then assume that the $k^th$ individual with the message at time $n$, passes it on to $A_k,n$ other individuals. We then have the following recursion for $X_n$:
$$X_n=sum_k=1^X_n-1A_k,n$$
Further assume that the random variables $A_k,n$ are independent and identically distributed for all $k$ and $n$. Use the notation $a$ and $v$ for the (common) mean and variance $A_k,n$. We assume $a > 1$ in all questions.
Calculate $E[X_n|X_n−1]$ and $E[X_2|X_n−1]$.
Calculate $E[X_n|X_0]$ and $E[X_2|X_0]$.
(tip: calculate first $E[X_2|X_n−2]$,
$E[X_2|X_n−3]dots$ before you try to get the
expression for $E[X_2|X_0]$)
Can anyone help me how to solve it when i dont have value for $X_0$ and $k$ starts from $1$?
stochastic-processes
edited Aug 7 at 21:34
mrtaurho
698219
asked Aug 7 at 18:43
New comer
6
3
Welcome to MSE! Your question will be much easier for you and everyone else to read (and therefore more likely to be answered) if you use MathJax to format the mathematics in it: math.meta.stackexchange.com/questions/5020/…
– Robert Howard
Aug 7 at 18:47
I have added the MathJax notation for you @New comer. Please check it to verify if everything is right like this. Further look at it and maybe this can help you to understand how it works.
– mrtaurho
Aug 7 at 20:01
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let $X_n$ be the number of individuals that have received the message at time $n$, then assume that the $k^th$ individual with the message at time $n$, passes it on to $A_k,n$ other individuals. We then have the following recursion for $X_n$:
$$X_n=sum_k=1^X_n-1A_k,n$$
Further assume that the random variables $A_k,n$ are independent and identically distributed for all $k$ and $n$. Use the notation $a$ and $v$ for the (common) mean and variance $A_k,n$. We assume $a > 1$ in all questions.
Calculate $E[X_n|X_n−1]$ and $E[X_2|X_n−1]$.
Calculate $E[X_n|X_0]$ and $E[X_2|X_0]$.
(tip: calculate first $E[X_2|X_n−2]$,
$E[X_2|X_n−3]dots$ before you try to get the
expression for $E[X_2|X_0]$)
Can anyone help me how to solve it when i dont have value for $X_0$ and $k$ starts from $1$?
stochastic-processes
edited Aug 7 at 21:34
mrtaurho
698219
asked Aug 7 at 18:43
New comer
6
We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let $X_n$ be the number of individuals that have received the message at time $n$, then assume that the $k^th$ individual with the message at time $n$, passes it on to $A_k,n$ other individuals. We then have the following recursion for $X_n$:
$$X_n=sum_k=1^X_n-1A_k,n$$
Further assume that the random variables $A_k,n$ are independent and identically distributed for all $k$ and $n$. Use the notation $a$ and $v$ for the (common) mean and variance $A_k,n$. We assume $a > 1$ in all questions.
Calculate $E[X_n|X_n−1]$ and $E[X_2|X_n−1]$.
Calculate $E[X_n|X_0]$ and $E[X_2|X_0]$.
(tip: calculate first $E[X_2|X_n−2]$,
$E[X_2|X_n−3]dots$ before you try to get the
expression for $E[X_2|X_0]$)
Can anyone help me how to solve it when i dont have value for $X_0$ and $k$ starts from $1$?
stochastic-processes
edited Aug 7 at 21:34
mrtaurho
698219
asked Aug 7 at 18:43
New comer
6
edited Aug 7 at 21:34
mrtaurho
698219
edited Aug 7 at 21:34
mrtaurho
698219
edited Aug 7 at 21:34
mrtaurho
698219
698219
asked Aug 7 at 18:43
New comer
6
asked Aug 7 at 18:43
New comer
6
asked Aug 7 at 18:43
New comer
6
6
3
Welcome to MSE! Your question will be much easier for you and everyone else to read (and therefore more likely to be answered) if you use MathJax to format the mathematics in it: math.meta.stackexchange.com/questions/5020/…
– Robert Howard
Aug 7 at 18:47
I have added the MathJax notation for you @New comer. Please check it to verify if everything is right like this. Further look at it and maybe this can help you to understand how it works.
– mrtaurho
Aug 7 at 20:01
add a comment |Â
3
Welcome to MSE! Your question will be much easier for you and everyone else to read (and therefore more likely to be answered) if you use MathJax to format the mathematics in it: math.meta.stackexchange.com/questions/5020/…
– Robert Howard
Aug 7 at 18:47
I have added the MathJax notation for you @New comer. Please check it to verify if everything is right like this. Further look at it and maybe this can help you to understand how it works.
– mrtaurho
Aug 7 at 20:01
3
3
Welcome to MSE! Your question will be much easier for you and everyone else to read (and therefore more likely to be answered) if you use MathJax to format the mathematics in it: math.meta.stackexchange.com/questions/5020/…
– Robert Howard
Aug 7 at 18:47
Welcome to MSE! Your question will be much easier for you and everyone else to read (and therefore more likely to be answered) if you use MathJax to format the mathematics in it: math.meta.stackexchange.com/questions/5020/…
– Robert Howard
Aug 7 at 18:47
I have added the MathJax notation for you @New comer. Please check it to verify if everything is right like this. Further look at it and maybe this can help you to understand how it works.
– mrtaurho
Aug 7 at 20:01
I have added the MathJax notation for you @New comer. Please check it to verify if everything is right like this. Further look at it and maybe this can help you to understand how it works.
– mrtaurho
Aug 7 at 20:01
add a comment |Â
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Welcome to MSE! Your question will be much easier for you and everyone else to read (and therefore more likely to be answered) if you use MathJax to format the mathematics in it: math.meta.stackexchange.com/questions/5020/…
– Robert Howard
Aug 7 at 18:47
I have added the MathJax notation for you @New comer. Please check it to verify if everything is right like this. Further look at it and maybe this can help you to understand how it works.
– mrtaurho
Aug 7 at 20:01