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If $N$ is independent of each $X_i$, is $N$ independent of $sum_i X_i$?
What is this operation on random variables called?Conditional probability given independent random variablesQuestion about definition of independent discrete random variables.If $N$ is an integral random variable, can $S_N$ be expressed in terms of the $S_n$ and $P(N=n)$?Find distribution of rv X_N where N is independent rv and each X_i~exp(lambda_i)Conditional Expectation of random sum of independent random variables (when $N$ and $X_i$ are dependent)Find the value of $mathbbE(X_1+X_2+ldots+X_N)$ of i.i.d random variables $X_i$s.Size of $Omega$ to have independent +1/-1 random variablesDistribution of the Number of Distinct Items in a Sequence of Independent Random VariablesMoment-generating function of $m$ independent variables
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Let $X_1,X_2, ...,X_n$ a sequence of discrete random variables i.i.d, and $N>0$ a positive discrete random variable, such that $N$ and $X_i$ are independent for $i=1,...,n$.
Can we say that $X_1+...+X_n$ and $N$ are independent?
probability probability-theory random-variables
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add a comment |
$begingroup$
Let $X_1,X_2, ...,X_n$ a sequence of discrete random variables i.i.d, and $N>0$ a positive discrete random variable, such that $N$ and $X_i$ are independent for $i=1,...,n$.
Can we say that $X_1+...+X_n$ and $N$ are independent?
probability probability-theory random-variables
$endgroup$
add a comment |
$begingroup$
Let $X_1,X_2, ...,X_n$ a sequence of discrete random variables i.i.d, and $N>0$ a positive discrete random variable, such that $N$ and $X_i$ are independent for $i=1,...,n$.
Can we say that $X_1+...+X_n$ and $N$ are independent?
probability probability-theory random-variables
$endgroup$
Let $X_1,X_2, ...,X_n$ a sequence of discrete random variables i.i.d, and $N>0$ a positive discrete random variable, such that $N$ and $X_i$ are independent for $i=1,...,n$.
Can we say that $X_1+...+X_n$ and $N$ are independent?
probability probability-theory random-variables
probability probability-theory random-variables
edited Mar 15 at 18:44
Mike Earnest
25.5k22151
25.5k22151
asked Mar 14 at 17:31
AnasAnas
254
254
add a comment |
add a comment |
1 Answer
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No. An example:
Let the $X_i$ each be the discrete uniform distribution on $0,frac1n,frac2n,dots,fracn-1n$ and let $N$ be the fractional part of $X_1+X_2+cdots+X_n$. It's independent of any one $X_i$, identically distributed to them, and completely determined by the sum.
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I think $X_i$ are meant to be discrete. Not much change in your example though.
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– Radost
Mar 14 at 17:38
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I noticed that after writing it. Already edited.
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– jmerry
Mar 14 at 17:39
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How do you justify that $N$ so defined is independent of the $X_i$ ?
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– Yves Daoust
Mar 14 at 17:41
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Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
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– Anas
Mar 14 at 17:45
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@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
|
show 2 more comments
Your Answer
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1 Answer
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1 Answer
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$begingroup$
No. An example:
Let the $X_i$ each be the discrete uniform distribution on $0,frac1n,frac2n,dots,fracn-1n$ and let $N$ be the fractional part of $X_1+X_2+cdots+X_n$. It's independent of any one $X_i$, identically distributed to them, and completely determined by the sum.
$endgroup$
$begingroup$
I think $X_i$ are meant to be discrete. Not much change in your example though.
$endgroup$
– Radost
Mar 14 at 17:38
$begingroup$
I noticed that after writing it. Already edited.
$endgroup$
– jmerry
Mar 14 at 17:39
$begingroup$
How do you justify that $N$ so defined is independent of the $X_i$ ?
$endgroup$
– Yves Daoust
Mar 14 at 17:41
$begingroup$
Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
$endgroup$
– Anas
Mar 14 at 17:45
$begingroup$
@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
|
show 2 more comments
$begingroup$
No. An example:
Let the $X_i$ each be the discrete uniform distribution on $0,frac1n,frac2n,dots,fracn-1n$ and let $N$ be the fractional part of $X_1+X_2+cdots+X_n$. It's independent of any one $X_i$, identically distributed to them, and completely determined by the sum.
$endgroup$
$begingroup$
I think $X_i$ are meant to be discrete. Not much change in your example though.
$endgroup$
– Radost
Mar 14 at 17:38
$begingroup$
I noticed that after writing it. Already edited.
$endgroup$
– jmerry
Mar 14 at 17:39
$begingroup$
How do you justify that $N$ so defined is independent of the $X_i$ ?
$endgroup$
– Yves Daoust
Mar 14 at 17:41
$begingroup$
Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
$endgroup$
– Anas
Mar 14 at 17:45
$begingroup$
@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
|
show 2 more comments
$begingroup$
No. An example:
Let the $X_i$ each be the discrete uniform distribution on $0,frac1n,frac2n,dots,fracn-1n$ and let $N$ be the fractional part of $X_1+X_2+cdots+X_n$. It's independent of any one $X_i$, identically distributed to them, and completely determined by the sum.
$endgroup$
No. An example:
Let the $X_i$ each be the discrete uniform distribution on $0,frac1n,frac2n,dots,fracn-1n$ and let $N$ be the fractional part of $X_1+X_2+cdots+X_n$. It's independent of any one $X_i$, identically distributed to them, and completely determined by the sum.
answered Mar 14 at 17:35
jmerryjmerry
15.3k1632
15.3k1632
$begingroup$
I think $X_i$ are meant to be discrete. Not much change in your example though.
$endgroup$
– Radost
Mar 14 at 17:38
$begingroup$
I noticed that after writing it. Already edited.
$endgroup$
– jmerry
Mar 14 at 17:39
$begingroup$
How do you justify that $N$ so defined is independent of the $X_i$ ?
$endgroup$
– Yves Daoust
Mar 14 at 17:41
$begingroup$
Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
$endgroup$
– Anas
Mar 14 at 17:45
$begingroup$
@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
|
show 2 more comments
$begingroup$
I think $X_i$ are meant to be discrete. Not much change in your example though.
$endgroup$
– Radost
Mar 14 at 17:38
$begingroup$
I noticed that after writing it. Already edited.
$endgroup$
– jmerry
Mar 14 at 17:39
$begingroup$
How do you justify that $N$ so defined is independent of the $X_i$ ?
$endgroup$
– Yves Daoust
Mar 14 at 17:41
$begingroup$
Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
$endgroup$
– Anas
Mar 14 at 17:45
$begingroup$
@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
$begingroup$
I think $X_i$ are meant to be discrete. Not much change in your example though.
$endgroup$
– Radost
Mar 14 at 17:38
$begingroup$
I think $X_i$ are meant to be discrete. Not much change in your example though.
$endgroup$
– Radost
Mar 14 at 17:38
$begingroup$
I noticed that after writing it. Already edited.
$endgroup$
– jmerry
Mar 14 at 17:39
$begingroup$
I noticed that after writing it. Already edited.
$endgroup$
– jmerry
Mar 14 at 17:39
$begingroup$
How do you justify that $N$ so defined is independent of the $X_i$ ?
$endgroup$
– Yves Daoust
Mar 14 at 17:41
$begingroup$
How do you justify that $N$ so defined is independent of the $X_i$ ?
$endgroup$
– Yves Daoust
Mar 14 at 17:41
$begingroup$
Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
$endgroup$
– Anas
Mar 14 at 17:45
$begingroup$
Can we say that $[N=n]$ is independent of $[X_1+...X_n=k]$ ?
$endgroup$
– Anas
Mar 14 at 17:45
$begingroup$
@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
$begingroup$
@YvesDaoust: Even if you lock down exactly what $X_1,X_2,dots,X_n-1$ are, adding another discrete uniform to that evens out the probabilities of the fractional part to $frac1n$ at each possible value. That's the independence.
$endgroup$
– jmerry
Mar 14 at 17:56
|
show 2 more comments
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