Mixing
If $ln(x)$ is gamma distributed, what is the distribution of $x$?
$begingroup$
Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
probability probability-distributions gamma-distribution
edited Jan 21 at 13:25
gt6989b
34.6k22456
asked Jan 21 at 4:23
rmb5769rmb5769
32
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add a comment |
$begingroup$
Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
probability probability-distributions gamma-distribution
edited Jan 21 at 13:25
gt6989b
34.6k22456
asked Jan 21 at 4:23
rmb5769rmb5769
32
$endgroup$
add a comment |
$begingroup$
Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
probability probability-distributions gamma-distribution
edited Jan 21 at 13:25
gt6989b
34.6k22456
asked Jan 21 at 4:23
rmb5769rmb5769
32
$endgroup$
Additionally, if someone could help calculate the mean and variance of $X$, that would be greatly appreciated.
probability probability-distributions gamma-distribution
probability probability-distributions gamma-distribution
edited Jan 21 at 13:25
gt6989b
34.6k22456
asked Jan 21 at 4:23
rmb5769rmb5769
32
edited Jan 21 at 13:25
gt6989b
34.6k22456
asked Jan 21 at 4:23
rmb5769rmb5769
32
edited Jan 21 at 13:25
gt6989b
34.6k22456
edited Jan 21 at 13:25
gt6989b
34.6k22456
edited Jan 21 at 13:25
gt6989b
34.6k22456
34.6k22456
asked Jan 21 at 4:23
rmb5769rmb5769
32
asked Jan 21 at 4:23
rmb5769rmb5769
32
asked Jan 21 at 4:23
rmb5769rmb5769
32
32
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
HINT
Let $ln X = Y sim Gamma(alpha, beta)$ then
$$
F_X(x)
= mathbb{P}[X le x]
= mathbb{P}[ln X le ln x]
= F_Y(ln x)
$$
and using the Chain Rule,
$$
f_X(x) = F'_X(x) = f_Y(ln x)/x
$$
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
$endgroup$
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
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
$begingroup$
HINT
Let $ln X = Y sim Gamma(alpha, beta)$ then
$$
F_X(x)
= mathbb{P}[X le x]
= mathbb{P}[ln X le ln x]
= F_Y(ln x)
$$
and using the Chain Rule,
$$
f_X(x) = F'_X(x) = f_Y(ln x)/x
$$
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
$endgroup$
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
add a comment |
$begingroup$
HINT
Let $ln X = Y sim Gamma(alpha, beta)$ then
$$
F_X(x)
= mathbb{P}[X le x]
= mathbb{P}[ln X le ln x]
= F_Y(ln x)
$$
and using the Chain Rule,
$$
f_X(x) = F'_X(x) = f_Y(ln x)/x
$$
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
$endgroup$
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
add a comment |
$begingroup$
HINT
Let $ln X = Y sim Gamma(alpha, beta)$ then
$$
F_X(x)
= mathbb{P}[X le x]
= mathbb{P}[ln X le ln x]
= F_Y(ln x)
$$
and using the Chain Rule,
$$
f_X(x) = F'_X(x) = f_Y(ln x)/x
$$
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
$endgroup$
HINT
Let $ln X = Y sim Gamma(alpha, beta)$ then
$$
F_X(x)
= mathbb{P}[X le x]
= mathbb{P}[ln X le ln x]
= F_Y(ln x)
$$
and using the Chain Rule,
$$
f_X(x) = F'_X(x) = f_Y(ln x)/x
$$
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
answered Jan 21 at 4:32
gt6989bgt6989b
34.6k22456
34.6k22456
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
add a comment |
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
Thanks for your help! Is there a name for this specific distribution?
$endgroup$
– rmb5769
Jan 21 at 6:37
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
$begingroup$
@rmb5769 not that i know of, unfortunately
$endgroup$
– gt6989b
Jan 21 at 13:25
add a comment |
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