  (Solution) 170B Probability Theory Puck Rombach HW 4, Due: 2/1/16 Problem 1 A Nonnegative Integer-valued Random Variable X Has One Of The Following Two... | Snapessays.com

(Solution) 170B Probability Theory Puck Rombach HW 4, due: 2/1/16 Problem 1 A nonnegative integer-valued random variable X has one of the following two...

A nonnegative integer-valued 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 integer-valued 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 4-sided, a 5-sided, a 6-sided, a 7-sided, and an 8-sided 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

.

.........

Solution details:
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