Page 257 - Probability and Statistical Inference

234 4. Functions of Random Variables and Sampling Distribution

One will recall from (1.7.27) that this pdf is known as the lognormal density
and the corresponding X is called a lognormal random variable. Suppose that
X , X are iid having the common pdf f(x). Let r and s be arbitrary, but fixed
1
2
real numbers. Then, find the pdf of the random variable {Hint: Does
taking log help?}
4.4.14 Suppose that X , X , X are iid Gamma(α, β), α > 0, β > 0. Define
3
2
1
U = X + X + X , U = X /(X + X ), and U = X /(X + X + X ). Solve the
2
3
3
3
1
2
1
2
2
1
1
3
2
following problems.
(i) Show that one can express x = u (1 – u )(1 – u ), x = u u (1 –
2
3
2
1
1
1 2
u ), x . Is this transformation one-to-one?
3
3
(ii) Determine the matrix J from (4.4.3) and show that
| det(J) | = ;
(iii) Start out with the joint pdf of X , X , X . Use the transformation
2
3
1
(x , x , x ) → (u , u , u ) to obtain the joint pdf of U , U , U ;
1 2 3 1 2 3 1 2 3
(iv) Argue that U , U , U are distributed independently. Show that
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2
1
(a) U is distributed as Gamma(3α, β), (b) U is distributed as
1
2
Beta(α, α), and (c) U is distributed as Beta(α, 2α). In this
3
part, recall the beta function and the Beta pdf from (1.6.25)-
(1.6.26) and (1.7.35) respectively.
4.4.15 Suppose that X , X , X , X are iid Gamma(α, β), α > 0, β > 0.
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1
2
4
Define U = X + X + X + X , U = X /(X + X ), U = X /(X + X + X ),
3
3
3
2
2
4
2
1
2
3
1
1
1
2
and U = X /(X + X + X + X ). Solve the following problems.
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2
4
3
4
4
(i) Show that one can express x = u (1 – u )(1 – u )(1 – u ), x =
2
4
2
1
1
3
u u (1 – u )(1 – u ), x = u u (1 – u ), x = u u . Is this transfor
4
1 3
1 2
3
4
1 4
3
4
mation one-to-one?
(ii) Determine the matrix J from (4.4.3) and show that | det(J)
| = u (1 – u )(1 – u )(1 – u );
3
2
1
4
(iii) Start out with the joint pdf of X , X , X , X . Use the transforma
2
3
1
4
tion (x , x , x , x ) →u , u , u , u ) to obtain the joint pdf of U ,
1
1
2
4
1
4
3
2
3
U , U , U ;
4
2
3
(iv) Argue that U , U , U , U are distributed independently Show
2
1
4
3
that (a) U is distributed as Gamma(4α, β), (b) U is distributed
1
2
as Beta(α, α), (c) U is distributed as Beta(α, 2α), and (d) U is
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4
Beta(α, 3α). Recall the beta function and Beta pdf from (1.6.25)-
(1.6.26) and (1.7.35) respectively.
4.4.16 (Example 4.4.13 Continued) Provide all the details in the Example
4.4.13.
4.4.17 (Example 4.4.14 Continued) Provide all the details in the Example
4.4.14.