Mixing
How to generate samples of Poisson-Lognormal distribution
$begingroup$
I would like to compute samples of the number of product purchased in a supermarket. I want to model it with a mixed Poisson lognormal distribution.
- Items purchased $x$ of a given consumer follow a Poisson distribution with the mean rate of purchasing $lambda$
$f(x)_{poisson}=e^{-lambda}lambda^{x}/x!$
- The mean rate of purchasing $lambda$ of different consumers differ, and are distributed lognormally.
By combining 1 and 2, the probability density function of $x$ items purchased results the following distribution:
$f(x)=frac{1}{x!sigmasqrt{2pi}}int_{0}^{infty}lambda^{x-1} e^{-lambda} e^{frac{(log(lambda-mu)^2}{2sigma^2} }text{d}lambda $
I am simulating purchases in a supermarket, and I would like to sample this distribution to know the items purchased by each consumer by computing samples of this random variable, for example, with mean 2.32 and standard deviation 1.29 of products.
My simulator can generate easily samples of a Poisson and a Gaussian random variable. Is there any form to compute samples of the above mixed distribution using random samples of Poisson and Gaussian random variables?.
I have seen in this post that lognormal and Gaussian distribution are related, but I do not know how to apply this to compute Poisson-lognormal random variable of items purchased.
normal-distribution sampling random-variable random-generation lognormal
asked Jan 14 at 12:20
user1993416user1993416
1334
$endgroup$
add a comment |
$begingroup$
I would like to compute samples of the number of product purchased in a supermarket. I want to model it with a mixed Poisson lognormal distribution.
- Items purchased $x$ of a given consumer follow a Poisson distribution with the mean rate of purchasing $lambda$
$f(x)_{poisson}=e^{-lambda}lambda^{x}/x!$
- The mean rate of purchasing $lambda$ of different consumers differ, and are distributed lognormally.
By combining 1 and 2, the probability density function of $x$ items purchased results the following distribution:
$f(x)=frac{1}{x!sigmasqrt{2pi}}int_{0}^{infty}lambda^{x-1} e^{-lambda} e^{frac{(log(lambda-mu)^2}{2sigma^2} }text{d}lambda $
I am simulating purchases in a supermarket, and I would like to sample this distribution to know the items purchased by each consumer by computing samples of this random variable, for example, with mean 2.32 and standard deviation 1.29 of products.
My simulator can generate easily samples of a Poisson and a Gaussian random variable. Is there any form to compute samples of the above mixed distribution using random samples of Poisson and Gaussian random variables?.
I have seen in this post that lognormal and Gaussian distribution are related, but I do not know how to apply this to compute Poisson-lognormal random variable of items purchased.
normal-distribution sampling random-variable random-generation lognormal
asked Jan 14 at 12:20
user1993416user1993416
1334
$endgroup$
add a comment |
$begingroup$
I would like to compute samples of the number of product purchased in a supermarket. I want to model it with a mixed Poisson lognormal distribution.
- Items purchased $x$ of a given consumer follow a Poisson distribution with the mean rate of purchasing $lambda$
$f(x)_{poisson}=e^{-lambda}lambda^{x}/x!$
- The mean rate of purchasing $lambda$ of different consumers differ, and are distributed lognormally.
By combining 1 and 2, the probability density function of $x$ items purchased results the following distribution:
$f(x)=frac{1}{x!sigmasqrt{2pi}}int_{0}^{infty}lambda^{x-1} e^{-lambda} e^{frac{(log(lambda-mu)^2}{2sigma^2} }text{d}lambda $
I am simulating purchases in a supermarket, and I would like to sample this distribution to know the items purchased by each consumer by computing samples of this random variable, for example, with mean 2.32 and standard deviation 1.29 of products.
My simulator can generate easily samples of a Poisson and a Gaussian random variable. Is there any form to compute samples of the above mixed distribution using random samples of Poisson and Gaussian random variables?.
I have seen in this post that lognormal and Gaussian distribution are related, but I do not know how to apply this to compute Poisson-lognormal random variable of items purchased.
normal-distribution sampling random-variable random-generation lognormal
asked Jan 14 at 12:20
user1993416user1993416
1334
$endgroup$
I would like to compute samples of the number of product purchased in a supermarket. I want to model it with a mixed Poisson lognormal distribution.
- Items purchased $x$ of a given consumer follow a Poisson distribution with the mean rate of purchasing $lambda$
$f(x)_{poisson}=e^{-lambda}lambda^{x}/x!$
- The mean rate of purchasing $lambda$ of different consumers differ, and are distributed lognormally.
By combining 1 and 2, the probability density function of $x$ items purchased results the following distribution:
$f(x)=frac{1}{x!sigmasqrt{2pi}}int_{0}^{infty}lambda^{x-1} e^{-lambda} e^{frac{(log(lambda-mu)^2}{2sigma^2} }text{d}lambda $
I am simulating purchases in a supermarket, and I would like to sample this distribution to know the items purchased by each consumer by computing samples of this random variable, for example, with mean 2.32 and standard deviation 1.29 of products.
My simulator can generate easily samples of a Poisson and a Gaussian random variable. Is there any form to compute samples of the above mixed distribution using random samples of Poisson and Gaussian random variables?.
I have seen in this post that lognormal and Gaussian distribution are related, but I do not know how to apply this to compute Poisson-lognormal random variable of items purchased.
normal-distribution sampling random-variable random-generation lognormal
normal-distribution sampling random-variable random-generation lognormal
asked Jan 14 at 12:20
user1993416user1993416
1334
asked Jan 14 at 12:20
user1993416user1993416
1334
edited Jan 16 at 12:02
user1993416
asked Jan 14 at 12:20
user1993416user1993416
1334
asked Jan 14 at 12:20
user1993416user1993416
1334
asked Jan 14 at 12:20
user1993416user1993416
1334
1334
add a comment |
add a comment |
1 Answer
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You can generate a sample by first generating a normally distributed value, then take the exponent of that, then use that as the parameter in a Poisson distribution and take a sample from that distribution. The resulting samples of this three-step process will be Poisson-Lognormally distributed.
answered Jan 14 at 12:26
GijsGijs
1,739714
$endgroup$
1
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
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$begingroup$
You can generate a sample by first generating a normally distributed value, then take the exponent of that, then use that as the parameter in a Poisson distribution and take a sample from that distribution. The resulting samples of this three-step process will be Poisson-Lognormally distributed.
answered Jan 14 at 12:26
GijsGijs
1,739714
$endgroup$
1
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
add a comment |
$begingroup$
You can generate a sample by first generating a normally distributed value, then take the exponent of that, then use that as the parameter in a Poisson distribution and take a sample from that distribution. The resulting samples of this three-step process will be Poisson-Lognormally distributed.
answered Jan 14 at 12:26
GijsGijs
1,739714
$endgroup$
1
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
add a comment |
$begingroup$
You can generate a sample by first generating a normally distributed value, then take the exponent of that, then use that as the parameter in a Poisson distribution and take a sample from that distribution. The resulting samples of this three-step process will be Poisson-Lognormally distributed.
answered Jan 14 at 12:26
GijsGijs
1,739714
$endgroup$
You can generate a sample by first generating a normally distributed value, then take the exponent of that, then use that as the parameter in a Poisson distribution and take a sample from that distribution. The resulting samples of this three-step process will be Poisson-Lognormally distributed.
answered Jan 14 at 12:26
GijsGijs
1,739714
answered Jan 14 at 12:26
GijsGijs
1,739714
answered Jan 14 at 12:26
GijsGijs
1,739714
answered Jan 14 at 12:26
GijsGijs
1,739714
1,739714
1
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
add a comment |
1
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
1
1
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
Jan 14 at 12:29
add a comment |
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