Quiz 6
MATH 3280 3.00
Page 2 of 2
Question 1 Prove that for two RVs
T
(
x
)
≥
0
, T
(
y
)
≥
0, we have
Cov(
T
(
x
)
, T
(
y
)) =
Z
∞
0
Z
∞
0
(
P
(
T
(
x
)
≥
u, T
(
y
)
≥
v
)
-
P
(
T
(
x
)
≥
u
)
P
(
T
(
y
)
≥
v
)) d
u
d
v,
given that the covariance is well-defined and finite. Use this identity to show that if
the RVs
T
(
x
)
≥
0
, T
(
y
)
≥
0 are PQD, that is positively quadrant dependent, then
these RVs are positively correlated.
Question 2 Let
T
0
(
x
)
, T
0
(
y
)
, ξ
be independent RVs distributed gamma with shape parameters
γ
0
x
, γ
0
y
, γ
, respectively, and rate parameter
λ
; all parameters are positive. Also, set
T
(
x
) =
T
0
(
x
) +
ξ
and
T
(
y
) =
T
0
(
y
) +
ξ
. Show that the RVs
T
(
x
) and
T
(
y
) are PQD.
Derive the covariance of the RVs
T
(
x
) and
T
(
y
).
The End.
