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EL 630: Homework 3
1.
A fair coin is tossed 4 times. The random variable
X
is defined as the total
number of heads. Write the events
{}
{}
{}
{}
.
5
;
2
;
2
;
3
=
<
≤
=
X
X
X
X
2.
Do the following functions define probability distribution functions?
(i)
<
≥

=

.
0
,
0
,
0
,
0
,
1
)
(
x
x
e
x
F
x
X
α
(ii)
<
≤
<
=
.
2
/
1
,
1
,
2
/
1
0
,
,
0
,
0
)
(
x
x
x
x
x
F
X
(iii)
)
(
tan
1
2
1
)
(
1
x
x
F
X

+
=
π
for
.
+∞
<
<
∞

x
3.
In the following cases determine if
)
(
x
f
X
is a valid probability density function
(p.d.f) for some suitable
K
.
If so find
K
. If not give reasons.
(i)
<
<

+
=
.
,
0
,
5
5
),
(
sin
1
)
(
10
otherwise
x
Kx
x
f
X
(ii)
(Discrete case)
( 29
.
,
2
,
1
,
0
,
1
0
,
=
<
<
=
=
k
p
Kp
k
X
P
k
(iii)
≥
=


.
,
0
,
0
,
)
(
2
/
)
(ln
2
otherwise
x
e
x
K
x
f
x
X
μ
Here
is a known constant and ln stands for natural log.
4.
Past statistics (collected over the last 100 years or more) show that out of every
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Unformatted text preview: 1000 cavelry men in the French army, one person has died of horse kick every year. Assuming that this phenomenon has an (approximate) Poisson distribution (this is indeed found to be true!) find the probability that of the present 10000 caverly men not more than 3 will have such an ending. 5. If the random variables X and Y are such that ( 29 ( 29 ξ Y X ≤ for every , then show that ) ( ) ( ϖ Y X F F ≥ for every . +∞ ≤ ≤ ∞...
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This note was uploaded on 04/24/2011 for the course ECE 6303 taught by Professor Voltz during the Spring '11 term at NYU Poly.
 Spring '11
 voltz

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