(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

 

.

 

.........

 


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