Homework Set No. 3 – Probability Theory (235A), Fall 2009
Posted: 10/13/09 — Due: 10/20/09
1.
Let
X
be an exponential r.v. with parameter
λ
, i.e.,
F
X
(
x
) = (1

e

λx
)1
[0
,
∞
)
(
x
). Define
random variables
Y
=
b
X
c
:= sup
{
n
∈
Z
:
n
≤
x
}
(“the integer part of
X
”)
,
Z
=
{
X
}
:=
X
 b
X
c
(“the fractional part of
X
”)
.
(a) Compute the (1dimensional) distributions of
Y
and
Z
(in the case of
Y
, since it’s a
discrete random variable it is most convenient to describe the distribution by giving the
individual probabilities
P
(
Y
=
n
)
, n
= 0
,
1
,
2
, . . .
; for
Z
one should compute either the
distribution function or density function).
(b) Show that
Y
and
Z
are independent. (Hint: Check that
P
(
Y
=
n, Z
≤
t
) =
P
(
Y
=
n
)
P
(
Z
≤
t
) for all
n
and
t
.)
2.
(a) Let
X, Y
be independent r.v.’s.
Define
U
= min(
X, Y
),
V
= max(
X, Y
).
Find
expressions for the distribution functions
F
U
and
F
V
in terms of the distribution functions
of
X
and
Y
.
(b) Assume that
X
∼
Exp(
λ
)
, Y
∼
Exp(
μ
) (and are independent as before). Prove that
min(
X, Y
) has distribution Exp(
λ
+
μ
). Try to give an intuitive explanation in terms of
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 Fall '11
 DanRomik
 Probability, Probability theory, homework set no., fX +Y, respective densities fX

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