Page 259 - Probability and Statistical Inference

236 4. Functions of Random Variables and Sampling Distribution

distributed. Show that the common pdf of X , X , X must be the normal pdf.
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2
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{Hint: Can the Remark 4.4.4 be used to solve this problem?}
4.5.1 In the Theorem 4.5.1, provide all the details in the derivation of the
pdf of t , the Student’s t with ν degrees of freedom.
í
4.5.2 Consider the pdf h(w) of t , the Student’s t with ν degrees of free-
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dom, for w ∈ ℜ. In the case ν = 1, show that h(w) reduces to the
Cauchy density.
4.5.3 Generalize the result given in the Example 4.5.2 in the case of the k-
sample problem with k(≥ 3).
4.5.4 With n(≥ 4) iid random variables from N(0, σ ), consider the Helmert
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variables Y , ..., Y from Example 4.4.9. Hence, obtain the distribution
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of and that of W ?
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4.5.5 Suppose that X , ..., X are iid random variables having a common
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pdf given by f(x) = σ exp{–(x – µ)/σ}I(x > µ) for – ∞ < µ < ∞, 0 < σ < ∞.
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Here, µ is the location parameter and ó is the scale parameter. Consider (4.4.18)
and let U = n(X – µ) / σ, V = 2T / σ.
n:1
r
(i) Find the pdf of U and E(U ) for all fixed numbers r > 0;
(ii) Evaluate E(V ) for all fixed real numbers r;
r
(iii) What is the distribution of W = σU ÷ [T/(n – 1)]?
{Caution: In parts (i)-(ii), watch for the condition on n as needed.}
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4.5.6 Suppose that X , ..., X are iid N(µ, σ ). Recall the Helmert variables
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Y , ..., Y from the Example 4.4.9.
1 n
(i) Find the pdf of Y /Y for i ≠ j = 2, ..., n;
j
i
(ii) Find the pdf of Y / |Y | for i ≠ j = 2, ..., n;
j
i
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2
(iii) Find the pdf of Y / Y for i ≠ j = 2, ..., n;
j
i
(iv) Find the pdf of Y / T where
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i
(v) Find the pdf of /U where
for i ≠ j = 2, ..., n.
{Hint: Recall the distributional properties of the Helmert variables.}
4.5.7 (Exercise 4.1.1 Continued) Let Z , Z be iid standard normal. From
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the Exercise 4.5.2, note that the random variable Z /Z , denoted by X, has
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the Cauchy pdf given in (1.7.31). Use this sampling distribution to evaluate
where b(> 0) is fixed but arbitrary. {Hint: Express the re-
quired probability as and then integrate the Cauchy pdf
within the appropriate interval. Does this match with the answer given for the
Exercise 4.1.1?}
4.5.8 (Exercise 4.5.7 Continued) Let Z , Z be iid standard normal.
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Evaluate where c is a fixed but arbitrary real number.