STAT 410
Summer 2009
Homework #3
(due Monday, June 29, by 4:00 p.m.)
1.
Suppose Heidi has a fair 4sided die, and Alex has a fair 6sided die.
Each day,
they roll their dice (independently) until someone rolls a “1”.
(Then the person
who did not roll a “1” does the dishes.)
Find the probability that …
a)
they roll the first “1” at the same time (after equal number of attempts);
b)
it takes Alex twice as many attempts as it does Heidi to roll the first “1”;
c)
Alex rolls the first “1” before Heidi does;
d)
Alex rolls the first “1” on an evennumbered attempt;
e)
Alex rolls a “1” before he rolls an even number.
2.
Let
X, Y, and Z
be
i.i.d.
Uniform
[
0
,
1
]
random variables
Find the probability
distribution of
W = X + Y + Z.
That is, find
(
w
f
W
.
Hint:
If
V = X + Y,
we know the
p.d.f.
of
V,
f
V
(
v
)
(
see Examples for 06/24/2009
)
:
f
V
(
v
)
=
v
if
0 <
v
< 1,
f
V
(
v
)
= 2 –
v
if
1 <
v
< 2,
f
V
(
v
)
= 0
otherwise.
Now use convolution formula to find the
p.d.f.
of
W = V + Z.
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 Spring '08
 STEPANOV
 Statistics, Normal Distribution, Probability distribution, Probability theory, probability density function

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