Proving the relation between probability and Laplace transform
$begingroup$
I need to prove that equation 2 is equivalent to 1:
begin{gather*}
sum_{i=1}^K λ_iint_{mathbb{R}^2} mathbb{P}left(frac{P_i h_{x_i} l(x_i) x_i}{I_{x_i} + σ^2} > β_iright) ,mathrm dx_i, tag{1}\
sum_{i=1}^K λ_iint_{mathbb R^2} mathcal{L}_{I_{x_i}}left(frac{β_i}{P_i l(x_i)}right)expleft(frac{-β_i σ^2}{P_i l(x_i) x_i}right), ,mathrm{d} x_i tag{2}
end{gather*}
where $mathcal{L}_{I_{x_i}}$ is Laplace of $I_{x_i}$ and $h_{x_i}$ has Rayleigh distribution.
Equation 1 represents the integral over Euclidean space for that the probability of received signal to noise ratio (SINR) over a multi-tier network is greater than predefined threshold $β_i$ for any $i$-th tier.
linear-algebra statistics
edited Dec 2 '18 at 11:10
Saad
19.7k92352
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
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add a comment |
$begingroup$
I need to prove that equation 2 is equivalent to 1:
begin{gather*}
sum_{i=1}^K λ_iint_{mathbb{R}^2} mathbb{P}left(frac{P_i h_{x_i} l(x_i) x_i}{I_{x_i} + σ^2} > β_iright) ,mathrm dx_i, tag{1}\
sum_{i=1}^K λ_iint_{mathbb R^2} mathcal{L}_{I_{x_i}}left(frac{β_i}{P_i l(x_i)}right)expleft(frac{-β_i σ^2}{P_i l(x_i) x_i}right), ,mathrm{d} x_i tag{2}
end{gather*}
where $mathcal{L}_{I_{x_i}}$ is Laplace of $I_{x_i}$ and $h_{x_i}$ has Rayleigh distribution.
Equation 1 represents the integral over Euclidean space for that the probability of received signal to noise ratio (SINR) over a multi-tier network is greater than predefined threshold $β_i$ for any $i$-th tier.
linear-algebra statistics
edited Dec 2 '18 at 11:10
Saad
19.7k92352
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
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2
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
add a comment |
$begingroup$
I need to prove that equation 2 is equivalent to 1:
begin{gather*}
sum_{i=1}^K λ_iint_{mathbb{R}^2} mathbb{P}left(frac{P_i h_{x_i} l(x_i) x_i}{I_{x_i} + σ^2} > β_iright) ,mathrm dx_i, tag{1}\
sum_{i=1}^K λ_iint_{mathbb R^2} mathcal{L}_{I_{x_i}}left(frac{β_i}{P_i l(x_i)}right)expleft(frac{-β_i σ^2}{P_i l(x_i) x_i}right), ,mathrm{d} x_i tag{2}
end{gather*}
where $mathcal{L}_{I_{x_i}}$ is Laplace of $I_{x_i}$ and $h_{x_i}$ has Rayleigh distribution.
Equation 1 represents the integral over Euclidean space for that the probability of received signal to noise ratio (SINR) over a multi-tier network is greater than predefined threshold $β_i$ for any $i$-th tier.
linear-algebra statistics
edited Dec 2 '18 at 11:10
Saad
19.7k92352
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
$endgroup$
I need to prove that equation 2 is equivalent to 1:
begin{gather*}
sum_{i=1}^K λ_iint_{mathbb{R}^2} mathbb{P}left(frac{P_i h_{x_i} l(x_i) x_i}{I_{x_i} + σ^2} > β_iright) ,mathrm dx_i, tag{1}\
sum_{i=1}^K λ_iint_{mathbb R^2} mathcal{L}_{I_{x_i}}left(frac{β_i}{P_i l(x_i)}right)expleft(frac{-β_i σ^2}{P_i l(x_i) x_i}right), ,mathrm{d} x_i tag{2}
end{gather*}
where $mathcal{L}_{I_{x_i}}$ is Laplace of $I_{x_i}$ and $h_{x_i}$ has Rayleigh distribution.
Equation 1 represents the integral over Euclidean space for that the probability of received signal to noise ratio (SINR) over a multi-tier network is greater than predefined threshold $β_i$ for any $i$-th tier.
linear-algebra statistics
linear-algebra statistics
edited Dec 2 '18 at 11:10
Saad
19.7k92352
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
edited Dec 2 '18 at 11:10
Saad
19.7k92352
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
edited Dec 2 '18 at 11:10
Saad
19.7k92352
edited Dec 2 '18 at 11:10
Saad
19.7k92352
edited Dec 2 '18 at 11:10
Saad
19.7k92352
19.7k92352
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
asked Dec 2 '18 at 10:29
Ahmed abd el azizAhmed abd el aziz
11
11
2
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
add a comment |
2
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
2
2
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
$begingroup$
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36
add a comment |
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
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– José Carlos Santos
Dec 2 '18 at 10:36