UCSD ECE153
Handout #15
Prof. YoungHan Kim
Thursday, April 21, 2011
Homework Set #4
Due: Thursday, April 28, 2010
1.
Two independent uniform random variables.
Let
X
and
Y
be independently and uni
formly drawn from the interval [0
,
1].
(a) Find the pdf of
U
= max(
X, Y
)
.
(b) Find the pdf of
V
= min(
X, Y
).
(c) Find the pdf of
W
=
U

V
.
(d) Find the probability
P
{
X

Y
 ≥
1
/
2
}
.
2.
Waiting time at the bank.
Consider a bank with two tellers. The service times for the
tellers are independent exponentially distributed random variables
X
1
∼
Exp(
λ
1
) and
X
2
∼
Exp(
λ
2
), respectively. You arrive at the bank and ±nd that both tellers are busy
but that nobody else is waiting to be served. You are served by the ±rst available teller
once he/she becomes free. Let the random variable
Y
denote your waiting time. Find
the pdf of
Y
.
3.
Two envelopes.
An amount
A
is placed in one envelope and the amount 2
A
is placed
in another envelope. The amount
A
is ±xed but unknown to you. The envelopes are
shu²ed and you are given one of the envelopes at random. Let
X
denote the amount
you observe in this envelope. Designate by
Y
the amount in the other envelope. Thus
(
X, Y
) =
b
(
A,
2
A
)
,
with probability
1
2
,
(2
A, A
)
,
with probability
1
2
.
You may keep the envelope you are given, or you can switch envelopes and receive the
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 Spring '08
 staff
 Variance, Probability theory, probability density function, uniform random variables

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