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MATH 180A – Introduction to Probability
Review for the Final (NOT due)
problems focus on what was covered since midterm 2.
1. Two balls are drawn with replacement from an urn containing
n
balls numbered 1
through
n
. Let
X
be the number the ﬁrst draw and
Y
the number on the second
draw. Compute the PDF of
Z
=
X
+
Y
. Repeat when the balls are drawn without
replacement.
2. A fair die is rolled repeatedly. Let
X
be the number of trials until all numbers 1
through 6 appear. For example,
X
= 11 in the following example:
1 4 2 2 5 6 2 1 1 4 3
2 2 2 5 4 1 4 4
Compute
E
(
X
). (Hint: Consider
X
i
, the number of trials required to get a new number
given that
i

1 diﬀerent numbers have already appeared. Express
X
in terms of the
X
i
’s and show that each
X
i
has a geometric distribution.)
3. Consider
X
and
Y
with the following joint PDF:
f
(
x, y
) =
C
(1

x

y
)
,
x, y
∈
(0
,
1)
,
x
+
y
≤
1
.
(a) Compute
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 Fall '08
 Castro
 Sets, Probability

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