CS206
HW4
October 20, 2009
(* due date is November 4, 2009)
1. A fair die is tossed twice. Let
X
= the sum of the faces,
Y
= the maximum of the two faces,
and
Z
=

face 1

face 2

.
(a) Write down the value of
X
,
Y
,
Z
, and
W
=
XZ
for each outcome
w
∈
S
.
(b) Find
Range
(
X
),
Range
(
Y
),
Range
(
Z
), and
Range
(
W
).
(c) Describe the partitions
A
X
and
A
Z
induced by these random variables.
(d) Find
f
X
,
f
Y
,
f
Z
, and
f
W
, the frequency functions.
(e) Are the events
A
=
{
w
∈
S
:
X
(
w
) = 7
}
and
B
=
{
w
∈
S
:
Z
(
w
) = 1
}
independent?
2. (*) A fair coin is tossed four times. Consider the following random variables on
S
, the sample
space:
X
= the number of Heads;
Y
= the length of the longest block of successive Tails (0
if NO Tails);
Z
= the number of the toss on which the last Tail occurred (0 if NO Tails);
W
= max(
X, Y
);
V
= min(
X, Z
).
(a) List the elements of
S
. For each
w
∈
S
, write the value of each random variable.
(b) Find the
Range
of each random variable.
(c) Find the frequency function of each random variable and plot it.
(d) Describe the partitions induced by
X
and
W
.
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 Spring '08
 Fredman
 Probability theory, Randomness, Probability space, frequency function

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