STAB52H3 Midterm TestSahITday,June 21, 2008

Duration: one hour and fifty minutes

(Sclwh"~ )Please do NOT open this booklet until you are asked to do so

There are 9 pages including this page. Please check to see if you have all the pages.

Aids allowed: This test is open book. You are allowed to use a non-programmablecalculator. No other aids are allowed.

All your work must be presented clearly in order to get credit. Just an answer with noother work shown will only qualify for zero credit. Show your work and answer in thespace provided, in ink. Pencil may be used, but then any re-grading will NOT beallowed.

You will get a mark out of 90 for this test.

Last name:

First name:

Other names:

Student number:

Circle your tutorial TUT001(Tue 12:00-13:00) TUT002(Tue 13:00-14:00)

Page 1 of9

1) A box contains 3 red balls, 4 green balls and 5 blue balls. Two balls are picked atrandom from this box (with replacement).

a)[4po7~=e~:~)ili:a;O~~~e~o~0 Uy g~p .~ :3 f. ~ -+ If 'f 4-- -t-S7'_.)..- -J 2-"1- f2-

:::- () .3'12-

b) [4 points] What is the probability that none of the two balls selected is blue.

PC~ ~ -t;k) ~ p (~C-&<:)}", 1.

c) [4 points] What is the probability that one (and only one) of the two balls selected isblue.

p (6~ ~ o:fJ ~i;k)::-- p (g bL crY (SC&J

.5:-;:1-+ '+:xr ::::O_¥6 I12)Z\2

Page2 of9

2. In a group of 100Find the . 'people~40 owna cat~25 ownaprobabIlIty that a person chosen at rand d°frg, and 1.5own a cat and a dog.

om om this grou

a) owns a cat or a dog [2' P

, pomts] n

pee):: iE. PO)::: ~~ pee ~ J))I D [) ) I DO )

rYc.(j{Sb) ::- p( C)-tPCP)~ pee J.D)~ Y:9 +~ -5- ~ ().S

b) owns a dog or a cat, but not both \f;:"nts] I c:>C> J DO

P(C oy) ~~ ~):::- PCCeND) - P Cc ~ j))

.- (!).s- - IS'-' -,,:)---- - u. 2:>~tOO

[3points]c) owns a dog, given that he owns a cat

()(DIL) ~ ~CDC)peL ).

IS / f 00

LPI v 0 D

..-

d) does not own a cat, given that he owns a dog. [3 points]

PCeL]))~

PC:DJP(])~C)-

.- PC]))-

...--

- l...l- -/ of)

PC])) - P(DC)

PGY)z~ --~ rroO I~::: 0,,+-

,00

Page 3 of9

3. Suppose that~ Yare llD geometric(O.l).

Find

a) P(X > 2)[3 points]

b) P(X>7 I X>5)[4 points]

--- p (;><';7 ::; ~ ;><;>s)-?(X >5)

P(X'7?-) - 1'J-- -

PCl<?S) -CZ-t

-

c) P(Y =2X) ::[5 points]

D6

Z p(;<,=L) ;"'2-i)l.::. \0

..-- 2"'" p(x ~ 2f(/= 2- l )L~D

L 2-l

~ CZ0<5 t I?' 'L tL :=-:0

-r-..--

F~-{ --p

-,

I - J 7~ -

.:: () ,v:s to { Page 4 of9

4. The number of customers entering a shop during a certain time period has a Poissondistribution with A =10. Each customer entering the shop makes a purchase withprobability 0.7 independently of other customers. Find the probability that there willbe no sales (Le.no one makes a purchase) in that time period. [10 points]

--iO V]

~ /V"Vi'I

-

-/0.Q-

3,Q.

-g-::::- 12---

Page5 of9

5) A, B and C are events satisfying the following conditions,A) A and B are independent.B) A and C are independentC) B and C are disjoint

Show that the events A and B u C are independent. [5 points] -)..,. '\ /

PC A (I ((bUC)) =- p(lt-£; u ft-() / - 'r Ie - th~ PC kg,") -t- P (IK ') - p ( At. n A'- ) . IS - r.

~o=- P{AJ P({?J i PCA)r (c) --- ~ - ( ~ / A1- g. A .1-~), }

=- p ( f\-J L p ( ~ ') -T PC ~ ")J ~

C " (7 ~L ~ ~IC1~=- PCA- 1 P (rbv L ') - - \ 0 ~ .

~.

(Q

6) If A, B and C are events such that PP(AuBIC)-P A (C»O,showthat- (IC)+P(BIC)-P(AnBIC) [5 points]

PCAV€,\L"::' p(Q,-v\?")nCJ~peC)

=- eC A-L.- V (SC)- PC~

-=. p ekL ') -\-P (~ C") - e( A-@, L '),pc C)

- PCA-L") ecg,c.J - PC1\{SLJ- -t ~ ..- -c') pc() rC~),

-::- p ( A\ L") -t p( (S1 c) - P C f\iS \c ) ,

Page6 of9

7) Suppose that the random variable Y has density function

{

cy2 1::;y ::;2f{y) = 0 elsewhere.

a)...f~ the value of c.

JJc()~ ~ \_C<>

b) Find P{Y > 1.5) [3 points]

c) Find P{Y > 1.51Y> 1.2) [4 points]

---

---

Page 7of9

8) X and Yare continuous random variables with the joint density function given by:

I(x,y) ={

4X2y + 2y5 O:$;x:$;I,O:$;y:$;1

o elsewhere.

-

9) LetXbe a random variable with p.d.£ given by:

Determine the p.d.f. of U =1- X2. [6 points]L.

~r/- u ~ hcr:-). ~ 1- A<;~{L '

h ' iLJ It5IJLr~ CfY\, (9<. LA cl

.~I "\.-- J (--- L1

h (0\ J-

,u.

{

3X2 for 0 < x < 1,

I(x) = 0 elsewhere.

-I

J;< (~ (L\ 'I ').~

\ ~((hIClA»))

-~ Jr-,,- ~2-

CI

Page 8 of9

10) Let X and Y have joint p.d.f. given by

{

-xf(x,y)= e ,

0,O~y~x<ooelsewhere.

a) Calculate P(X ~ 2Y). [7 points]

-

b) Find P(Y < 11

X - 2) [8'

]-. pomtr

p (y ~ \ \"{-::- ~ ) ::: J 1-C 'j \ 2-) ~11)< I

'X. ~ob

(I

J-~ ~ -~

j ('It) =- ~ ~. I = /l-Q.-/< 0 0 ~ ~( 06

- - -

-'A1L- -71Xs2-

- \-YL

J C~'\~)Ylj J

J>l1'\ )

fiX:::7\. rV

'II X :0l- AJ LA ~', r l () \ 2- J .

~ p( y .( \ \ ~ ~"') =- ~1--'Page 9 of 9