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Homework Set No. 3 – Probability Theory (235A), Fall 2011
Due: Tuesday 10/18/11 at discussion section
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

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