Name:

PID:

Seat Number:

To a�rm your commitment to academic integrity,

write out the phrase “I excel with integrity”

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MATH 180A (B00) EXAM 2 November 20, 2019

INSTRUCTIONS — READ THIS NOW

• Write your name, PID, and current seat number, and write out the

academic integrity pledge in the indicated space above. Also, write your

name and PID on every page. Your score will not be recorded if

any of this identifying information or the pledge is missing.

• Write within the boxed areas, or your work will not be graded. If you

need more space, raise your hand. To receive full credit, your answers

must be neatly written and logically organized.

• To ask questions during the exam, remain in your seat and raise your

hand. Please show your ID to a proctor when you hand in your exam.

• You may not speak to any other student in the exam room while the

exam is in progress. You may not share any information about the

exam with any student who has not yet taken it.

• Turn o↵ and put away all cellphones, calculators, and other electronic

devices. You may not access any electronic device during the exam

period. You may use your one double-sided page of hand-written notes,

but no other books, notes, or aids.

• This is a 50-minute exam in design, but you have up to 2 hours to

complete it. There are 5 problems on 5 pages, worth a total of 50 points.

Read all the problems first before you start working on any of them, so

you can manage your time wisely.

• Have fun!

SolutionsAdI

Name: PID: (P. 3)

1. San Diego gets on average 1 major rain storm per year (defined as a single storm event with

more than 5mm of rain at the coast).

(a) (5 points) Using an appropriate Poisson distribution, compute the probability that San Diego

will get at least 3 major rain storms this year. (For reference in calculation: e�1 ⇡ 0.368.)

(b) (5 points) Let X be the number of major rain storms San Diego gets in a year. Show that the

probability calculated in part (a) is also equal to P(|X � 1| � 2). What does Chebyshev’s

inequality say about this probability for a general random variable X? How does it compare

to the exact answer you computed above?

Xs # storms I yearn Poisson4)

PIX 333=1 - PIK's )-

- l - E. e' ' II=L - e-

' ( lilt 's)=L -2.551=-84

.

IPIIX-1132) =P ( X- I > 2. or X- IS -2 )'- pl X - I >2) t 1PM - Is -2 )

f = PIX 337 tPkXbf soBad =L

,Var LXII ( Poisson 113 )

i. Chebyshev : IPC Ix- 1152)'

- ft '

Actual value 's 84<254:

Name: PID: (P. 5)

2. A poll is conducted to determine if UCSD students are in favor of the university administration’s

decision to spend billions of dollars on fancy new buildings instead of spending money hiring more

faculty to enable smaller class sizes. 400 students are polled, and the proportion who support the

administration’s plans is p̂ = 0.21.

(a) (5 points) Determine the 95% confidence interval for this poll. (The normal-table-lookup

you need is �(1.96) = 0.975.)

(a) (5 points) The poll organizers want to get a margin of error no more than ✏ = 0.02, withprobability 95%, in their next poll. How many students do they need to poll?

Pl Ip -pls e) 32$12erp ) - I > o.eswant

h=4eo

i. Want Io ( 2Effed 30.975want 2e . 2e > Io

- 'tests ) -- 1.96Want E3 t.no = 0.049

i. GA conf. int . e Cau- e. 049,auto

.

o491= ( o. 161

,0.259)

Now n Unknown,E -- o. 02 .

Want Io ( 2e rn ) 30.975

it.26oz) rn ? Io -40.9751 4.96

rn 311=49e. e 4

Se want h > 492=2401

Name: PID: (P. 7)

3. Based on past experience, we expect that the number of students who turn in tonight’s exam

between 9pm and 9:10pm will be about 20.

(a) (5 points) What is the probability that exactly 15 students will turn in the exam between

9pm and 9:10pm?

(b) (5 points) What is the probability that at least one student will turn in their exam between

9pm and 9:04pm?

-

X# students n Poisson l H.

Find X.

* Elt) = 20.

S. Pixels) -- E" 44, I 5.16%

T s time until first handed in r ExpH'

)Finelli

.

ECT) = ' ly'-i.

' t.se z

t

20 in lo minutes ⇒ average wait time : leg min =LSo

P ( at least ene by hole) = PITS4)= ! Ye

-Hdt

=- e-HI ! = I - e-

8e. 971

Name: PID: (P. 9)

4. Let U ⇠ Unif[0, 1] be a uniform random variable, and let X = 2 ln(1/U).

(a) (5 points) Compute the probability density function fX(x) for all x 2 R.

(b) (5 points) Given that X > 15, what is the probability that X > 17?

Start w Fx Itt -- RIX St) - Plzlnlyu) St )= Mfs eth)" t

:{:*:c: ' "in.io:*,Iet's too .

= , . go f e-the can't happene-th f e-the 6,11 It > o)I

-

f fth > , (tco)Ie. x - Ept 'd is Is

. .

Memoryless property of Expl 's ) :

Minnix " 37!= I - a- e-4)= f

'I 0.368

Name: PID: (P. 11)

5. Let X be a random variable with pdf fX(x) = xe�x {x � 0}.

(a) (7 points) Compute the moment generating function MX(t) = E(etX) of X. Make sure to

specify the domain on which it is finite. Also: explain how you can tell from the formula you

derive for MX(t) that fX really is a probability density function.

(b) (3 points) Is the distribution of X the only probability distribution that has the moment

generating function you computed in part (a)? Explain your answer.

Mdt) s Let's . Iet" . .co' da = !: e

"da

exponential blew upr? Mxltlzos

-f f > 1.

F t- I > o.

If td, { guetttdjjdr-x.f.se

" -"" I:-[¥t7dm= e - o - l , ett -015 )= ¥02 .

From here,we see

.

If,hilda = Mx le) = =/ ✓

Yes,Mx uniquely determines the dot .

of X,b/c My A) ca fer t in an interval

containing e ( namely L- es, D.)

.