EE 464, Mitra
Spring 2009
Homework 3
Due Friday, February 13, 2009, Box 8/EEB 104 9AM
This problem set investigates the definition of a random variable; the cumulative distribution
function and its properties; probability mass functions and probability density functions.
1. Write out the probabilities of the events
{
X < a
}
,
{
X
≤
a
}
,
{
a
≤
X < b
}
,
{
a
≤
X
≤
b
}
,
and
{
a < X < b
}
in terms of
F
X
(
x
)
and
P
[
X
=
x
]
, for
x
=
a, b
.
2. Show that the conditional distribution of
X
given the event
A
=
{
b < X
≤
a
}
is given by
F
X
(
x

A
)
=
0
x
≤
b
F
X
(
x
)

F
X
(
b
)
F
x
(
a
)

F
X
(
b
)
b < x
≤
a
1
x > a
NOTE: first determine an expression for
F
X
(
x

A
)
using the definition of conditional prob
ability, then evaluate the particular distribution function for the particular definition of
A
.
3. The probability mass function (pmf) for the Poisson random variable is given by
P
[
X
=
k
]
=
e

λ
λ
k
k
!
(a) For the values
λ
= 0
.
1
,
1
,
10
, plot the pmf. Comment on how the pmf changes as a
function of
λ
.
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 Spring '06
 Caire
 Normal Distribution, Gate, Probability theory, probability density function, Cumulative distribution function, CDF

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