MATH180A: Introduction to Probability

www.math.ucsd.edu/~ynemish/180a

Today: ASV 3.3, practice exam

Next: Midterm 1

Friday: ASV 3.3-3.4This week : regrade HW #2

+omorrow 8am - 11pm

midterm I

no homework

warm-upexercisehetxber.ir.that gives the member you get after

rolling a fair die once . Compute ECX).

If X discret,

then E (X ) = ¥ K - PCX -- K )

E-(X ) = t.PH-- 1) t 2. PCX = 2) t - - - t 6 P (X-- 6)

= t - ¥ t 2 . f- t - - - t 6. f-= f- (tt 2 t -- t 6) = ¥ . 6 = 3.5

Expectationofcontinuousr.v.isDet

.

1-et X be a continuons rv. with p .d-f. fx

.

The exportation of X is defined by+A

E(X) = fx - fxlx) dx--

Remords.Exportation is also colleta man

.

Exempte X - Umf [ a. b).ECX) - ? EN)=a¥

fx "" > {÷ ! s.tfxfxixsdx-labxb.adx-II.ca,= atb2

E-xpectationofcontinuousr.v.Examp-STakexr.ir.from the warm -

up exercée .

. " " )

p.at . f-* { % . * ,ECX) - ?

0 ,X ) (

EH)Ïxf¥dx= {xzp ,_⇒* = , - { = §%) "" ) ait

-

Not all r.v.is have welt - defined expectatives ,

Let X be a discret r.ir .

taKing values znforeuenn ,n > , }

Is this a welt-def.proba

- z"

for odd n

PLX-it-z.ph/=-zJ--E

MATH 180A - INTRODUCTION TO PROBABILITYPRACTICE MIDTERM #1

FALL 2019

Name (Last, First):

Student ID:

REMEMBER THIS EXAM IS GRADED BY AHUMAN BEING. WRITE YOUR SOLUTIONSNEATLY AND COHERENTLY, OR THEY RISKNOT RECEIVING FULL CREDIT.

THIS EXAM WILL BE SCANNED. MAKE SUREYOU WRITE ALL SOLUTIONS ON THE PAPERPROVIDED. DO NOT REMOVE ANY OF THEPAGES.

1

2 FALL 2019

1. Let (⌦,F ,P) be a probability space.

(a) Suppose that A,B 2 F satisfy

P(A) + P(B) > 1.

Making no further assumptions on A and B, prove that A \ B 6= ;.

(b) Prove that A is independent from itself if and only if P(A) 2 {0, 1}.

Sol.I : Assume that AMB =D

.Then

p( AUB) =P (A) + PCB) > 1,contradiction .

AMB # 0

-

Sol . z : PCAUB) = PCAITPIB ) - PCAM B) Et

p (AMB) z PIA ) + PCB)- 130

⇒ AMB # $

A indep . of A if PLA! A) = PCA)- PCA )

PCA)

(PLA))! PIA) - o ⇒ PLAIE f0.13

MATH 180A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM #1 3

2. Roll two fair dice repeatedly. If the sum is � 10, then you win.

(a) What is the probability that you start by winning 3 times in a row?

(b) What is the probability that after rolling the pair of dice 5 times you win exactly 3

times?

(c) What is the probability that the first time you win is before the tenth roll (of the

pair), but after the fifth?

p=P (Success)=P ( roll 2 dire , get I to )

= %# { (4,6 ) , 15,5) , (6/4) , (5/6) , ( 6,5 ) , (6,6 ) }

=t6

XRBfs.to ) gives # ofsuccesses after 3 games

PCX -- 3) = (f) psa.pt ¥

Yu 13f51 f) gives # successes in 5 games

PIY - 3) = (E) psa-pj-IO.fi =

Z - Geom (f) describes first winning game

PG ez c- 9) =P (E-6) xp (7--7) + PA -_ 8) tpfz -- G)

=p-pip + ( t-pip + A- M'pt ( i -pip= (tp Ip ( It ( t - plt ( t - pit ( i -pt )

- t -MX = t - pt - t -n"

4 FALL 2019

3. A box contains 3 coins, two of which are fair and the third has probability 3/4 of coming

up heads. A coin is chosen randomly from the box and tossed 3 times.

(a) What is the probability that all 3 tosses are heads?

(b) Given that all three tosses are heads, what is the probability that the biased coin

was chosen?

A-- { three heads ) B ,> { chosen coin is fair f

PLAIB ,)= #P Bz > { chose n cois isbiased }

PIAIBZ) > (E )?

p(B.) = } PIB, )= }

P (A) = PCAIBDP (B.) t PIAIBZ ) PCBL)

-- t - E + H )?t= . - -

=

192

PIBAIA) - MApf¥M=È) =± 43

192

MATH 180A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM #1 5

4. Let X be a discrete random variable taking the values {1, 2, . . . , n} all with equal prob-

ability. Let Y be another discrete random variable taking values in {1, 2, . . . , n}. Assume

that X and Y are independent. Show that P(X = Y ) =1n . (Hint: you do not need to know

the distribution of Y to calculate this.)

Play ) -_ ÊPHYIY --

k ) - Park )

" p(× -- Y , y - K ) - PCY - k )= -2-

K- t PCY - k )< In

can Star there nn

→ = ¥,PCX-x.tk/--,ZPCX--HPCY--k)C--l--tnE.,PH--kt=tn

6 FALL 2019

5. Consider a point P = (X, Y ) chosen uniformly at random inside of the triangle in R2that

has vertices (1, 0), (0, 1), and (0, 0). Let Z = max(X, Y ) be the random variable defined

as the maximum of the two coordinates of the point. For example, if P = (12 ,

13), then

Z = max(X, Y ) =12 . Determine the cumulative distribution function of Z. Determine if

Z is a continuous random variable, a discrete random variable, or neither. If continuous,

determine the probability density function of Z. If discrete, determine the probability mass

function of Z. If neither, explain why.

(Hint: Draw a picture.)

ÏËËË#÷ ËËË:÷:÷÷÷.EH ±

¥et

| - z ( t - s)'

, ¥ c 1

| ,SI 1

f fz f- I E = t - alt - { f- tz ,X continue»

Https :4 - 45 , { c Ss I

O i Ss l

MATH 180A - INTRODUCTION TO PROBABILITYPRACTICE MIDTERM #1 ADDITIONAL PROBLEMS

FALL 2019

Name (Last, First):

Student ID:

REMEMBER THIS EXAM IS GRADED BY AHUMAN BEING. WRITE YOUR SOLUTIONSNEATLY AND COHERENTLY, OR THEY RISKNOT RECEIVING FULL CREDIT.

THIS EXAM WILL BE SCANNED. MAKE SUREYOU WRITE ALL SOLUTIONS ON THE PAPERPROVIDED. DO NOT REMOVE ANY OF THEPAGES.

1

2 FALL 2019

1. At a political meeting there are 7 progressives and 7 conservatives. We choose five peopleuniformly at random to form a committee (president, vice-president and 3 regular members).

(a) Let A be the event that we end up with more conservatives than progressives. Whatis the probability of A?

(b) Let B be the event that Ronald, representing conservatives, becomes the president,and Felix, representing liberals, becomes the vice-president. What is the probabilityof B?

(a) Ai = { i conservative s in the comm iHee }

Plait =f- )

B = { more conservative s than progressives 4

B = As U Au UAs

PCB) =⑦E) + (7) f) + (E) (f)

(b) c = { Ronald president , Felix vice - president }

Pfc) =(E)← régaler members

== :*

✓ =#possible groups of 5 peoplewith

Ronald president and Felixvice president

We can compute it as1) chose 5 people out of

14,then chose among

those 5 pres . and vice-pres

( Ets - a-- au =

.

2) Chose pres . then vice pres , then3 other members

14.13 - (E) ="" ÎLE

3)Choo se 3 members ,then (from remaininq) pres and v - p : (f) II. 10 =- Itto =

MATH 180A - INTRODUCTION TO PROBABILITYPRACTICE MIDTERM #1 ADDITIONAL PROBLEMS3

2. Let A,B be events in a probability space (⌦,F ,P).(a) Suppose that A,B satisfy

P(A) + P(B) > 1.

Making no further assumptions on A and B, prove that A \ B 6= ;.(b) Suppose that A,B satisfy A\B = ;. If A and B are independent, what can you say

about P(A) and P(B)?(c) Suppose that P(A) = 0.5 and P(B) = 0.8. What possible range of values can P(A\B)

have?

(a) Similor as Ila ) in Practice midterm

(b) 1f A and B independent , then

PLAN B) = PCA)P(B) =P(d)= 0

⇒ either PCA) - 0 or PCB) - o

(c) L'c) OEPCAU B) Et

OEPCAU B) =P (A) + PCB)- PCAMB) Et

⇒ PCANB) ? PIA) + PCB)- 1=0.3

(ii ) AMBCA ,AMBCB

⇒ PLAN B) EPCA)> 0.5

,Pan B) EPCB) - 0.8

(c) , Cii ) ⇒ 0.3 EPCAMB) f0.5

4 FALL 2019

3. Suppose that we have two possibly unfair coins: the first coin takes heads with probabilityp 2 (0, 1) and tails with probability 1 � p; the second coin takes heads with probabilityq 2 (0, 1) and tails with probability 1 � q. The coins are independent of each other andconsecutive flips are also independent.

Consider the following game. Flip both coins simultaneously. If the coins land on thesame side (for example, both land on heads), then you win. Otherwise, then you lose. LetX be the number of times you play the game until you win. For example, X is equal to 1 ifyou win on your first play of the game. Determine the probability mass function of X.

P ( Win ) = PYH.HN + Pf (TT ) ) = pq + ( t - p) ( t - q )

= 1+2 pq- p - q = : s

Xr Geum ( s )

PCX -- K) = d- s)" 's = Cptq - zpqÏlttzpq - p - q )

MATH 180A - INTRODUCTION TO PROBABILITYPRACTICE MIDTERM #1 ADDITIONAL PROBLEMS5

4. Consider a point P = (X, Y ) chosen uniformly at random inside of the unit square[0, 1]2 = [0, 1] ⇥ [0, 1] = {(x, y) : 0 x, y 1}. Let Z = min(X, Y ) be the random variabledefined as the minimum of the two coordinates of the point. For example, if P = (12 ,

13),

then Z = min(12 ,13) =

13 . Determine the cumulative distribution function of Z. Determine

if Z is a continuous random variable, a discrete random variable, or neither. If continuous,determine the probability density function of Z. If discrete, determine the probability massfunction of Z. If neither, explain why.

(Hint: Draw a picture.)

¥ ::÷:÷"

* ÷÷÷

6 FALL 2019

5. You shoot an arrow (uniformly at random) at a round target of radius 50 cm. If you hita point at a distance 10 cm from the center of the target, you are awarded 10 points; ifthe point you hit is between 10 and 20 cm from the center, you get 5 point; if the point youhit is between 20 and 30 cm from the center, you get 3 points; if the point you hit is between30 and 40 points from the center you get 1 point; if you hit a point which is 40 cm or morefrom the center you get 0 points. Let X be a random variable that gives the total numberof points you get after two shots.

(a) Is the random variable X continuous, discrete, neither or both?(b) If X is continuous or discrete, compute the p.m.f./p.d.f. of X.(c) Compute and plot the c.d.f. of the random variable X.

-amateurs Ionest

(a) X is discrète ,since it has any fini le member

of possible outcomes

(b)

p.m.fi#E-:-"

¥ : ÷: ⑧¥ ,

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FX (s) = ¥ ,3 ES < 5

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