MA/STAT 519: Introduction to Probability Fall 2018, Mid-Term …stindel/teaching/ma416/... · 2019. 10. 10. · MA/STAT 519: Introduction to Probability Fall 2018, Mid-Term Examination

MA/STAT 519: Introduction to Probability

Fall 2018, Mid-Term Examination

Instructor: Yip

• This test booklet has FOUR QUESTIONS, totaling 80 points for the whole test. You have

75 minutes to do this test. Plan your time well. Read the questions carefully.

• This test is closed book, with no electronic device. One two-sided-8× 11 formula sheet

is allowed.

• In order to get full credits, you need to give correct and simplified answers and explain in

a comprehensible way how you arrive at them.

Name: (Department: )

Question Score

1.(20 pts)

2.(20 pts)

3.(20 pts)

4.(20 pts)

Total (80 pts)

1

Answerkey

1. Let X and Y be two independent discrete random variables. For each of the following cases,

compute the conditional distribution of X given X + Y , i.e. find

P(

X = i∣

∣

∣X + Y = j

)

If possible, relate the conditional distribution to some common, i.e. well known, distribution.

(a) X is Poisson with parameter λ and Y is Poisson with parameter µ.

(b) X is Binomial with parameter n and p and Y is Binomial with parameter m and p.

(c) X and Y are Geometric with parameter p.

2

PCX.it tYjJ P X i XtYj1PXtY jJPlX i Y ji7PlXtYjp X i PH jP XtY j

a Xv Poisson Y PoissonfmlYn PoissonHtml

t xii i s t.is ic si

i

BinCj.fII

You can use this blank page.

3

Bin Imp r.Binlm.plismIals.Y

sb 2HY Bin fntm.pl

Tiffany Hypergeometric

Cc Xngeomcp Yu geomlp

HY Neg Bink p

814814

i of Pfuniform doesnotdependoni

2. Suppose n balls are distributed at random into r boxes in such a way that each ball chooses

a box independently of each other. Let S be the number of empty boxes. Compute ES and

V ar(S).

(Hint: Consider the random variables Xi (for i = 1, 2, . . . , r) which equals 1 if the i-th box is

empty and 0 otherwise. Related S and the Xi’s.)

5

f Xt Xu Xr

ES EX Ext E Xr

EX 2x PIX D to xp o

PALED 14154 no A ballsBox List no ofboxes total no ofempty tochoose from boxes

Hence Ef r n

EH E E XiE Xi t jXiXjEl Xi gEHiXj

You can use this blank page.

6

Since Xi o Y Xi Xinoofboxes toEND E Xi choosefrom

Exit P Xi hXj D no ofmy ballstotal no ofpox i I are emptyboxes

Hence E rfrfjntrlr.it r fnVouCSJ

EC5 fESJ

rfr rYi rcr DfrIY EIHT

3. McDonald’s newest promotion is putting a toy inside every one of its hamburger. Suppose

there are N distinct types of toys and each of them is equally likely to be put inside any of

the hamburger. What is the expected value and variance of the number of hamburgers you

need to order (or eat) before you have a complete set of the N toys.

(Hint: consider the number of hamburgers you need to order (or eat) in between getting one

and two dinstinct types of toys, two and three distinct types of toys, and so forth.)

8

yrTF1 T2 Tz

T burgers

71 1

TegeomfftTs geomfEN

tag az

EIe.iYxLetS

Totalno.ofburgers purchasedThen 5 7 31 1 TN

You can use this blank page.

9

EfgeomIp ValGeomlpD opt

ES ET t EE t t ETNIt tf Yzx NyN fix It in

VanB Vat Var th t Vatu9since the Ti's are independent

o i t FIT

ftp.i f piHI

N Tiff

4. Consider a box with M white balls and N black balls. You are asked to get n balls (at

random) out of the box without replacement. Let X be the number of white balls obtained.

(a) Find the probability distribution of X.

(b) Find the expectation and variance of X.

(Hint: imagine that you get the balls sequentially, one by one. Introduce the indicator

functions Xi = 1 for i = 1, . . . n defined as Xi equals one if i-th ball is white and zero

otherwise. Relate X and the Xi’s.)

(c) Now suppose M and N tends to infinity such that MM+N

−→ p. Derive and identify the

limiting probability distribution of X.

(d) Under the same limiting procedure, find the limiting expectation and variance of X.

11

sN k

Iget n balls

Piti Miyazaki Hypergeometric

b X At Kt Xn

EX EX Ext tEXn

You can use this blank page.

12

E Xk PIX memoofwhiteballs

MtNr total no ofballsHence the ith djthEX MEX nM_ balls are white

Mtn

E I EXT t.EE iXjPlXi D IZ.gPlXi hXj D

mFntiEicm'InfinityTuffy mln DMCMMTN MTN t

vaixt ECXD CEXJ

nmfntntym7MY g taking

You can use this blank page.

13

cos Plait 7 Fi1

MIM 1 M ite NIN DIN D IN ostiaT.cn

MtN fMtN nt

I

jMCM D fatia n'terms

NIN D IN n in e terms

jMtN MtNntinterns

H hht cnn.inI fmMiniti x

H H InfinitiM N xw

f p 7 piqn imm arspB.ir n p

mNTN si p f

Note The answer also makes intuitive senseas M N to it does not makeany difference whether it is with orwithout replacement

s d Hence

EX np Vai k npofThe above can also becheckeddirectlyfrom the formula

EX nM_Man np

vaixt.mn o nfmInYimmnI CmFnT

n p t n in 1 p2 ripnp np npftp.J npq