Page 258 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 235

4.4.18 (Example 4.4.16 Continued) Suppose that (X , X ) is distributed as
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N (0, 0, σ , σ , ρ) with 0 < σ < ∞, –1 < ρ < 1. Define U = X /X and V = X .
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(i) Find the joint pdf of U and V;
(ii) In part (i), integrate out V and derive the marginal pdf of U;
(iii) When ρ = 0, the pdf in part (ii) coincides with the standard
Cauchy pdf, namely, π (1 + u ) for –∞ < u < ∞
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–1
(iv) When ρ ≠ 0, the pdf in part (ii) coincides with the Cauchy pdf
with appropriate location and scale depending on ρ.
4.4.19 Suppose that (X , X ) has the following joint pdf:
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Find the pdf of U = X – X . {Hint: Use the transformation u = x – x and v
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= x .}
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4.4.20 Suppose that (X , X ) has the following joint pdf:
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Show that the pdf of is given by

4.4.21 Suppose that (X , X ) has the following joint pdf:
1 2

Let U = X + X and V = X X .
1 2 1 2

(i) Show that the pdf of U is a(u)

(ii)Show that the pdf of V is b(v)

4.4.22 Suppose that X , X are iid random variables. We are told that X +
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X and X – X are independently distributed. Show that the common pdf of
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X , X must be same as the normal density. {Hint: Can the Remark 4.4.4 be
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used to solve this problem?}
4.4.23 Suppose that X , X , X are three iid random variables. We
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are also told that X + X + X and (X – X , X – X ) are independently
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