Statistics-relationships between gamma and exponential distribution
Just want to clarify whether the following is correct:
If gamma(a,b) ,then exp(a/b)?
where a,b are parameters for gamma
and a/b is the parameter for exp
for example, gamma(1,2)=exp(1/2)
Is this true for every a,b>0?
Thank you!
statistics probability-distributions
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
asked Mar 10 '16 at 16:25
UnusualSkill
442312
add a comment |
Just want to clarify whether the following is correct:
If gamma(a,b) ,then exp(a/b)?
where a,b are parameters for gamma
and a/b is the parameter for exp
for example, gamma(1,2)=exp(1/2)
Is this true for every a,b>0?
Thank you!
statistics probability-distributions
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
asked Mar 10 '16 at 16:25
UnusualSkill
442312
add a comment |
Just want to clarify whether the following is correct:
If gamma(a,b) ,then exp(a/b)?
where a,b are parameters for gamma
and a/b is the parameter for exp
for example, gamma(1,2)=exp(1/2)
Is this true for every a,b>0?
Thank you!
statistics probability-distributions
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
asked Mar 10 '16 at 16:25
UnusualSkill
442312
Just want to clarify whether the following is correct:
If gamma(a,b) ,then exp(a/b)?
where a,b are parameters for gamma
and a/b is the parameter for exp
for example, gamma(1,2)=exp(1/2)
Is this true for every a,b>0?
Thank you!
statistics probability-distributions
statistics probability-distributions
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
asked Mar 10 '16 at 16:25
UnusualSkill
442312
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
asked Mar 10 '16 at 16:25
UnusualSkill
442312
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
edited Nov 28 '18 at 0:09
carmichael561
46.9k54382
46.9k54382
asked Mar 10 '16 at 16:25
UnusualSkill
442312
asked Mar 10 '16 at 16:25
UnusualSkill
442312
asked Mar 10 '16 at 16:25
UnusualSkill
442312
442312
add a comment |
add a comment |
1 Answer
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The PDF for the $Gamma(alpha,lambda)$ distribution is
$$ f(x)=frac{x^{alpha-1}lambda^{alpha}}{Gamma(alpha)}e^{-lambda x}$$
for $x>0$. This is not the PDF for any exponential distribution unless $alpha=1$.
However, the gamma and exponential distributions are closely related: if $X_1,dots,X_k$ are independent and exponentially distributed with parameter $lambda$, then $X_1+dots+X_k$ is $Gamma$ distributed with parameters $k$ and $lambda$.
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
The PDF for the $Gamma(alpha,lambda)$ distribution is
$$ f(x)=frac{x^{alpha-1}lambda^{alpha}}{Gamma(alpha)}e^{-lambda x}$$
for $x>0$. This is not the PDF for any exponential distribution unless $alpha=1$.
However, the gamma and exponential distributions are closely related: if $X_1,dots,X_k$ are independent and exponentially distributed with parameter $lambda$, then $X_1+dots+X_k$ is $Gamma$ distributed with parameters $k$ and $lambda$.
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
add a comment |
The PDF for the $Gamma(alpha,lambda)$ distribution is
$$ f(x)=frac{x^{alpha-1}lambda^{alpha}}{Gamma(alpha)}e^{-lambda x}$$
for $x>0$. This is not the PDF for any exponential distribution unless $alpha=1$.
However, the gamma and exponential distributions are closely related: if $X_1,dots,X_k$ are independent and exponentially distributed with parameter $lambda$, then $X_1+dots+X_k$ is $Gamma$ distributed with parameters $k$ and $lambda$.
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
add a comment |
The PDF for the $Gamma(alpha,lambda)$ distribution is
$$ f(x)=frac{x^{alpha-1}lambda^{alpha}}{Gamma(alpha)}e^{-lambda x}$$
for $x>0$. This is not the PDF for any exponential distribution unless $alpha=1$.
However, the gamma and exponential distributions are closely related: if $X_1,dots,X_k$ are independent and exponentially distributed with parameter $lambda$, then $X_1+dots+X_k$ is $Gamma$ distributed with parameters $k$ and $lambda$.
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
The PDF for the $Gamma(alpha,lambda)$ distribution is
$$ f(x)=frac{x^{alpha-1}lambda^{alpha}}{Gamma(alpha)}e^{-lambda x}$$
for $x>0$. This is not the PDF for any exponential distribution unless $alpha=1$.
However, the gamma and exponential distributions are closely related: if $X_1,dots,X_k$ are independent and exponentially distributed with parameter $lambda$, then $X_1+dots+X_k$ is $Gamma$ distributed with parameters $k$ and $lambda$.
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
answered Mar 10 '16 at 16:38
carmichael561
46.9k54382
46.9k54382
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
add a comment |
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
means if alpha=1, then gamma(1,b)=exp(1/b)?
– UnusualSkill
Mar 10 '16 at 16:40
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
If $alpha=1$, then the gamma distribution is in fact an exponential distribution.
– carmichael561
Mar 10 '16 at 17:04
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
What if K isn’t an integer?
– John Cataldo
Feb 20 '18 at 13:51
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
@Stanislas Hildebrandt: the gamma distribution is defined for non-integer $alpha$, but it doesn't really make sense to sum up a non-integer number of exponential random variables.
– carmichael561
Feb 20 '18 at 16:18
add a comment |
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