Page 195 - Probability and Statistical Inference

172 3. Multivariate Random Variables

3.7.6 (Example 3.7.3 Continued) The zero correlation between two ran-
dom variables sometimes does indicate independence without bivariate nor-
mality. The Example 3.7.3 had dealt with a situation like this. In that example,
will the same conclusion hold if both X and X were allowed to take two
1
2
arbitrary values other than 0 and 1? {Hint: First recall from the Theorem
3.4.2, part (i) that the correlation coefficient between X and X would be the
2
1
same as that between Y and Y where we let Y = (X – ab , with a ∈ ℜ, b ∈
i
1
i
i
i i
i
2
ℜ being any fixed numbers. Then, use this result to reduce the given problem
+
to a situation similar to the one in the Example 3.7.3.}
3.7.7 Suppose that X and X have the joint pdf
1 2
where k is some positive constant. Show that X and X are uncorrelated, but
2
1
these are dependent random variables.
3.7.8 Suppose that X and X have the joint pdf
1 2
for –∞ < x , x < ∞. Show that X and X are uncorrelated, but these are
1
2
2
1
dependent random variables.
3.7.9 (Example 3.7.4 Continued) Suppose that (U , U ) is distributed as
2
1
N (5, 15, 8, 8, ρ) for some ρ ∈ (–1, 1). Let X = U + U and X = U – U .
1
1
1
2
2
2
2
Show that X and X are uncorrelated.
1 2
3.8.1 Verify the entries given in the Table 3.8.1.
3.8.2 Consider the Beta(α, β) distribution defined by (1.7.35). Does the
pdf belong to the appropriate (that is, one-or two-parameter) exponential family
when
(i) α is known, but β is unknown?
(ii) β is known, but α is unknown?
(iii) α and β are both unknown?
3.8.3 Consider the Beta(α, β) distribution defined by (1.7.35). Does the
pdf belong to the appropriate (that is, one- or two-parameter) exponential
family when
(i) α = β = θ, but θ(> 0) is unknown?
(ii) α = θ, β = 2θ, but θ(> 0) is unknown?
3.8.4 Suppose that X has the uniform distribution on the interval (–θ, θ)
with θ(> 0) unknown. Show that the corresponding pdf does not belong to