4.93 Give a simple method to simulate from the gamma distribution with parameters a and when the shape parameter a is equal to the integer n. Hint: The sum of n independent random variables, each having an exponential density with parameter A, has an Erlang density with parameters n and λ.
4.93 Give a simple method to simulate from the gamma distribution with parameters a and when the shape parameter a is equal to the integer n. Hint: The sum of n independent random variables, each having an exponential density with parameter A, has an Erlang density with parameters n and λ.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
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Question number 4.93
![09
On
od
C-
e
W.
is
re
in
re
te
-al
es
for s> 0, where c> 0 is a known constant. Determine the
ms of the molecule,
probability density of the kinetic energy E =
where m is the mass of one gas molecule.
4.90 Let the random variable X have uniform density on (-1,1). What is the
probability density of Y = In(|X") for any constant a > 0?
4.91 Let the variable Y be defined by Y 10%, where X is uniformly
distributed on (0, 1). Verify that Y has the density g(y)
Pindi for
1 < y < 10 and g(y) 0 otherwise. This is the so-called Benford
1
=
y
(10
density.
4.92 Verify that a random observation from the Weibull distribution with
shape parameter a and scale parameter λ can be simulated by taking
X = [-In(1 – U)]¹/a, where U is a random number from the interval
(0, 1). Also, use the inverse-transformation method to simulate a random
observation from the logistic distribution function F(x) = */(1+e) on
(-∞, ∞).
Jhon
list liw
to
4.93 Give a simple method to simulate from the gamma distribution with
parameters a and when the shape parameter a is equal to the integer
n. Hint: The sum of n independent random variables, each having
an exponential density with parameter λ, has an Erlang density with
parameters n and λ.
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Transcribed Image Text:09
On
od
C-
e
W.
is
re
in
re
te
-al
es
for s> 0, where c> 0 is a known constant. Determine the
ms of the molecule,
probability density of the kinetic energy E =
where m is the mass of one gas molecule.
4.90 Let the random variable X have uniform density on (-1,1). What is the
probability density of Y = In(|X") for any constant a > 0?
4.91 Let the variable Y be defined by Y 10%, where X is uniformly
distributed on (0, 1). Verify that Y has the density g(y)
Pindi for
1 < y < 10 and g(y) 0 otherwise. This is the so-called Benford
1
=
y
(10
density.
4.92 Verify that a random observation from the Weibull distribution with
shape parameter a and scale parameter λ can be simulated by taking
X = [-In(1 – U)]¹/a, where U is a random number from the interval
(0, 1). Also, use the inverse-transformation method to simulate a random
observation from the logistic distribution function F(x) = */(1+e) on
(-∞, ∞).
Jhon
list liw
to
4.93 Give a simple method to simulate from the gamma distribution with
parameters a and when the shape parameter a is equal to the integer
n. Hint: The sum of n independent random variables, each having
an exponential density with parameter λ, has an Erlang density with
parameters n and λ.
=
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