How to generate samples of Poisson-Lognormal distribution
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I would like to compute samples of the number of product purchased in a supermarket. In "Fundamental patterns of in-store shopper behavior" paper models 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
edited 8 hours ago
Stephan Kolassa
43.9k692161
asked 12 hours ago
user1993416user1993416
1284
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$begingroup$
I would like to compute samples of the number of product purchased in a supermarket. In "Fundamental patterns of in-store shopper behavior" paper models 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
edited 8 hours ago
Stephan Kolassa
43.9k692161
asked 12 hours ago
user1993416user1993416
1284
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I would like to compute samples of the number of product purchased in a supermarket. In "Fundamental patterns of in-store shopper behavior" paper models 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
edited 8 hours ago
Stephan Kolassa
43.9k692161
asked 12 hours ago
user1993416user1993416
1284
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I would like to compute samples of the number of product purchased in a supermarket. In "Fundamental patterns of in-store shopper behavior" paper models 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
edited 8 hours ago
Stephan Kolassa
43.9k692161
asked 12 hours ago
user1993416user1993416
1284
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 hours ago
Stephan Kolassa
43.9k692161
asked 12 hours ago
user1993416user1993416
1284
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 hours ago
Stephan Kolassa
43.9k692161
edited 8 hours ago
Stephan Kolassa
43.9k692161
edited 8 hours ago
Stephan Kolassa
43.9k692161
43.9k692161
asked 12 hours ago
user1993416user1993416
1284
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 12 hours ago
user1993416user1993416
1284
asked 12 hours ago
user1993416user1993416
1284
1284
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
user1993416 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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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 12 hours ago
GijsGijs
1,689613
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thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
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1 Answer
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1 Answer
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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 12 hours ago
GijsGijs
1,689613
$endgroup$
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
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 12 hours ago
GijsGijs
1,689613
$endgroup$
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
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 12 hours ago
GijsGijs
1,689613
$endgroup$
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 12 hours ago
GijsGijs
1,689613
answered 12 hours ago
GijsGijs
1,689613
answered 12 hours ago
GijsGijs
1,689613
answered 12 hours ago
GijsGijs
1,689613
1,689613
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
add a comment |
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
$begingroup$
thank you very much, I have to try this, seems simple process.
$endgroup$
– user1993416
12 hours ago
add a comment |
user1993416 is a new contributor. Be nice, and check out our Code of Conduct.
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