A nonnegative integervalued random variable X has one of the following two expressionsas its transform: 1. M(s) = e2(ees?1?1), 2. M(s)=e2(ees?1). (a) Explain why one of the two cannot possibly be the transform. (b) Use the true transform to find P(X = 0). this is the first question in the attached pdf. 170B Probability Theory
Puck Rombach
HW 4, due: 2
/
1
/
16
Problem 1
A nonnegative integervalued random variable
X
has one of the following two expressionsas its
transform:
1.
M
(
s
)
=
e
2(
e
e
s

1

1)
,
2.
M
(
s
)
=
e
2(
e
e
s

1)
.
(a)
Explain why one of the two cannot possibly be the transform.
(b)
Use the true transform to ?nd
P
(
X
=
0).
.........
Problem 2
Let
X
be a random variable with transform
M
X
(
s
) for all
s
. Use Markov’s Inequality to show that
P
(
X
?
x
)
?
e
log(
M
X
(1))

x
.
.........
Problem 3
We are going to use a moment generating function to answer a di
?
cult combinatorial question very
e
?
ciently. Suppose that we roll a 4sided, a 5sided, a 6sided, a 7sided, and an 8sided die, all at
the same time. Let
Y
be the sum of the 5 dice. Let
e
s
=
x
for simplicity. Try entering each of the
following lines into Wolfram Alpha (one at a time):
(x+xˆ2+xˆ3+xˆ4)/4*(x+xˆ2+xˆ3+xˆ4+xˆ5)/5
second derivative (eˆs+e(2*s)+e(3*s)+e(4*s)++e(5*s))/5, s=0
(a)
What is
P
(
Y
=
20)?
(b)
Find the expected value of
Y
3
+
Y
2
+
Y
.
.........
This question was answered on: May 23, 2022
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