Asymptotic growth of translation in random walk.
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Let's consider a random walk with translation $$F(N)=sum_{n=1}^{N}X(n)$$
Where $X(n)$ is a random variable distribiuted independently and equal to $0$ or $1$.
What is asymptotic growth of $F(N)$?
i.e.
I am looking for function $f$ which satisfy:$$F(N)=O(f(N))$$
Is this true that $f(N)=N^{0.5}$
I hope that someone will help me.
asymptotics random-walk
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add a comment |
$begingroup$
Let's consider a random walk with translation $$F(N)=sum_{n=1}^{N}X(n)$$
Where $X(n)$ is a random variable distribiuted independently and equal to $0$ or $1$.
What is asymptotic growth of $F(N)$?
i.e.
I am looking for function $f$ which satisfy:$$F(N)=O(f(N))$$
Is this true that $f(N)=N^{0.5}$
I hope that someone will help me.
asymptotics random-walk
$endgroup$
$begingroup$
To be clear, $X$ has a probability of $1/2$ being $0$ and $1/2$ being $1$, correct? It's not really clear from your wording.
$endgroup$
– Noble Mushtak
Dec 28 '18 at 21:35
add a comment |
$begingroup$
Let's consider a random walk with translation $$F(N)=sum_{n=1}^{N}X(n)$$
Where $X(n)$ is a random variable distribiuted independently and equal to $0$ or $1$.
What is asymptotic growth of $F(N)$?
i.e.
I am looking for function $f$ which satisfy:$$F(N)=O(f(N))$$
Is this true that $f(N)=N^{0.5}$
I hope that someone will help me.
asymptotics random-walk
$endgroup$
Let's consider a random walk with translation $$F(N)=sum_{n=1}^{N}X(n)$$
Where $X(n)$ is a random variable distribiuted independently and equal to $0$ or $1$.
What is asymptotic growth of $F(N)$?
i.e.
I am looking for function $f$ which satisfy:$$F(N)=O(f(N))$$
Is this true that $f(N)=N^{0.5}$
I hope that someone will help me.
asymptotics random-walk
asymptotics random-walk
asked Dec 28 '18 at 21:30
mkultramkultra
708
708
$begingroup$
To be clear, $X$ has a probability of $1/2$ being $0$ and $1/2$ being $1$, correct? It's not really clear from your wording.
$endgroup$
– Noble Mushtak
Dec 28 '18 at 21:35
add a comment |
$begingroup$
To be clear, $X$ has a probability of $1/2$ being $0$ and $1/2$ being $1$, correct? It's not really clear from your wording.
$endgroup$
– Noble Mushtak
Dec 28 '18 at 21:35
$begingroup$
To be clear, $X$ has a probability of $1/2$ being $0$ and $1/2$ being $1$, correct? It's not really clear from your wording.
$endgroup$
– Noble Mushtak
Dec 28 '18 at 21:35
$begingroup$
To be clear, $X$ has a probability of $1/2$ being $0$ and $1/2$ being $1$, correct? It's not really clear from your wording.
$endgroup$
– Noble Mushtak
Dec 28 '18 at 21:35
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
First, notice that $mu_X=frac 1 2$ and $sigma_X=frac 1 2(0-frac 1 2)^2+frac 1 2(1-frac 1 2)^2=frac 1 4$.
Let's consider:
$$G(N)=frac{F(N)}{N}=frac 1 N sum_{n=1}^N X(n)$$
By the Central Limit Theorem, as $Nrightarrow infty $, $G$ becomes normally distributed with $mu_G=mu_X=frac 1 2$ and $sigma_G=frac{sigma_X}{sqrt N}=frac{1}{4sqrt N}$
Now, $F=NG$. Therefore, as $Nrightarrow infty$, $F$ becomes normally distributed with $mu_F=Nmu_G=frac N 2$ and $sigma_F=Nsigma_G=frac{sqrt N}{4}$. Therefore, the expected value of $F$ is $O(N)$ and the standard deviation of $F$ is $O(sqrt N)$.
In short, in the average-case scenario, $F$ is $O(N)$. Also, in the worst-case scenario, $F=N$, so $F$ is $O(N)$ in worst-case scenario as well.
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But how the expected value determines "big O" notation?
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– mkultra
Dec 29 '18 at 12:20
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@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
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– Noble Mushtak
Dec 29 '18 at 15:09
add a comment |
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1 Answer
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1 Answer
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$begingroup$
First, notice that $mu_X=frac 1 2$ and $sigma_X=frac 1 2(0-frac 1 2)^2+frac 1 2(1-frac 1 2)^2=frac 1 4$.
Let's consider:
$$G(N)=frac{F(N)}{N}=frac 1 N sum_{n=1}^N X(n)$$
By the Central Limit Theorem, as $Nrightarrow infty $, $G$ becomes normally distributed with $mu_G=mu_X=frac 1 2$ and $sigma_G=frac{sigma_X}{sqrt N}=frac{1}{4sqrt N}$
Now, $F=NG$. Therefore, as $Nrightarrow infty$, $F$ becomes normally distributed with $mu_F=Nmu_G=frac N 2$ and $sigma_F=Nsigma_G=frac{sqrt N}{4}$. Therefore, the expected value of $F$ is $O(N)$ and the standard deviation of $F$ is $O(sqrt N)$.
In short, in the average-case scenario, $F$ is $O(N)$. Also, in the worst-case scenario, $F=N$, so $F$ is $O(N)$ in worst-case scenario as well.
$endgroup$
$begingroup$
But how the expected value determines "big O" notation?
$endgroup$
– mkultra
Dec 29 '18 at 12:20
$begingroup$
@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
$endgroup$
– Noble Mushtak
Dec 29 '18 at 15:09
add a comment |
$begingroup$
First, notice that $mu_X=frac 1 2$ and $sigma_X=frac 1 2(0-frac 1 2)^2+frac 1 2(1-frac 1 2)^2=frac 1 4$.
Let's consider:
$$G(N)=frac{F(N)}{N}=frac 1 N sum_{n=1}^N X(n)$$
By the Central Limit Theorem, as $Nrightarrow infty $, $G$ becomes normally distributed with $mu_G=mu_X=frac 1 2$ and $sigma_G=frac{sigma_X}{sqrt N}=frac{1}{4sqrt N}$
Now, $F=NG$. Therefore, as $Nrightarrow infty$, $F$ becomes normally distributed with $mu_F=Nmu_G=frac N 2$ and $sigma_F=Nsigma_G=frac{sqrt N}{4}$. Therefore, the expected value of $F$ is $O(N)$ and the standard deviation of $F$ is $O(sqrt N)$.
In short, in the average-case scenario, $F$ is $O(N)$. Also, in the worst-case scenario, $F=N$, so $F$ is $O(N)$ in worst-case scenario as well.
$endgroup$
$begingroup$
But how the expected value determines "big O" notation?
$endgroup$
– mkultra
Dec 29 '18 at 12:20
$begingroup$
@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
$endgroup$
– Noble Mushtak
Dec 29 '18 at 15:09
add a comment |
$begingroup$
First, notice that $mu_X=frac 1 2$ and $sigma_X=frac 1 2(0-frac 1 2)^2+frac 1 2(1-frac 1 2)^2=frac 1 4$.
Let's consider:
$$G(N)=frac{F(N)}{N}=frac 1 N sum_{n=1}^N X(n)$$
By the Central Limit Theorem, as $Nrightarrow infty $, $G$ becomes normally distributed with $mu_G=mu_X=frac 1 2$ and $sigma_G=frac{sigma_X}{sqrt N}=frac{1}{4sqrt N}$
Now, $F=NG$. Therefore, as $Nrightarrow infty$, $F$ becomes normally distributed with $mu_F=Nmu_G=frac N 2$ and $sigma_F=Nsigma_G=frac{sqrt N}{4}$. Therefore, the expected value of $F$ is $O(N)$ and the standard deviation of $F$ is $O(sqrt N)$.
In short, in the average-case scenario, $F$ is $O(N)$. Also, in the worst-case scenario, $F=N$, so $F$ is $O(N)$ in worst-case scenario as well.
$endgroup$
First, notice that $mu_X=frac 1 2$ and $sigma_X=frac 1 2(0-frac 1 2)^2+frac 1 2(1-frac 1 2)^2=frac 1 4$.
Let's consider:
$$G(N)=frac{F(N)}{N}=frac 1 N sum_{n=1}^N X(n)$$
By the Central Limit Theorem, as $Nrightarrow infty $, $G$ becomes normally distributed with $mu_G=mu_X=frac 1 2$ and $sigma_G=frac{sigma_X}{sqrt N}=frac{1}{4sqrt N}$
Now, $F=NG$. Therefore, as $Nrightarrow infty$, $F$ becomes normally distributed with $mu_F=Nmu_G=frac N 2$ and $sigma_F=Nsigma_G=frac{sqrt N}{4}$. Therefore, the expected value of $F$ is $O(N)$ and the standard deviation of $F$ is $O(sqrt N)$.
In short, in the average-case scenario, $F$ is $O(N)$. Also, in the worst-case scenario, $F=N$, so $F$ is $O(N)$ in worst-case scenario as well.
answered Dec 28 '18 at 21:43
Noble MushtakNoble Mushtak
15.3k1835
15.3k1835
$begingroup$
But how the expected value determines "big O" notation?
$endgroup$
– mkultra
Dec 29 '18 at 12:20
$begingroup$
@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
$endgroup$
– Noble Mushtak
Dec 29 '18 at 15:09
add a comment |
$begingroup$
But how the expected value determines "big O" notation?
$endgroup$
– mkultra
Dec 29 '18 at 12:20
$begingroup$
@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
$endgroup$
– Noble Mushtak
Dec 29 '18 at 15:09
$begingroup$
But how the expected value determines "big O" notation?
$endgroup$
– mkultra
Dec 29 '18 at 12:20
$begingroup$
But how the expected value determines "big O" notation?
$endgroup$
– mkultra
Dec 29 '18 at 12:20
$begingroup$
@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
$endgroup$
– Noble Mushtak
Dec 29 '18 at 15:09
$begingroup$
@mkultra Well, really the worst-case scenario determines big O notation, but sometimes, you will see people talk about average-case complexity as well: en.wikipedia.org/wiki/Average-case_complexity
$endgroup$
– Noble Mushtak
Dec 29 '18 at 15:09
add a comment |
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$begingroup$
To be clear, $X$ has a probability of $1/2$ being $0$ and $1/2$ being $1$, correct? It's not really clear from your wording.
$endgroup$
– Noble Mushtak
Dec 28 '18 at 21:35