an 5)people.stat.sc.edu/wang528/stat 511/hw 1 solutions.pdfhw 1-1 (due aug. 25, 2016) name: problem...
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HW 1-1 (Due Aug. 25, 2016) Name:
Problem 1. Suppose that A and B are two events. Write expressions involving unions, intersections,
and complements that describe the following:
1. Both events occur
2. At least one occurs
3. Neither occurs
4. Exactly one occurs
1
ANB
AUBAnd
#( An 5) U ( Bna )
Problem 2. suppose a family contains two children of di↵erent ages, and we are interested in the
gender of these children. Let F denote that a child is female and M that the child is male and let
a pair such as FM denote that the older child is female and the younger is male. THere are four
points in the set S of possible observations:
S = {FF, FM,MF,MM}.
Let A denote the subset of possibilities containing no males; B, the subset containing two males;
and C, the subset containing at least one male. List the elements of A, B, C, A [B, A \B, A [C,
A \ C, B [ C, B \ C, and C \ B̄.
2
A={ FF } B={ MM }
C= ( FM ,MF
,MM }
AUB= { FFMM }
AnB=¢Avc=S Ancaf
BVEC BAC=B
cnD= { FM ,MF }
Problem 3. Define the sequence of sets Aj = (1� 1/j, 2 + 1/j), for j = 1, 2, . . . . Then what are
[1j=1Aj and \1
j=1 Aj?
3
¥,Aj = ¥
,Ctjt ,
ztjt ) = ( 0,3 )
f),
Aj = [ 1,2 ]
HW 1-2 (Due Aug. 25, 2016) Name:
Problem 1. If A and B are two sets, draw Venn diagrams to verify the followings:
1. A = (A \B) [ (A \ ¯B)
2. if B ⇢ A then A = B [ (A \ ¯B)
Use the identities A = A \ S and S = B [ ¯B and a distributive law to prove that (mathematically,
not graphically)
1. A = (A \B) [ (A \ ¯B)
2. if B ⇢ A then A = B [ (A \ ¯B)
3. Further, show that (A \ B) and (A \ ¯B) are mutually exclusive and therefore that A is the
union of two mutually exclusive sets, (A \B) and (A \ ¯B).
4. Also show that B and (A \ ¯B) are mutually exclusive and if B ⇢ A, A is the union of two
mutually exclusive sets, B and (A \ ¯B).
1
"
K¥979 .ee#anit
l . CAABIUCANB )=AACBvBT=AnS=A
2 If BCA AAB=B
So from 1:
A=(ArB)UCAdB )
= BUCANB )
3. (AMDALAAD )=CAnA)n( BAD )=An¢=¢
4.
BACAAB )= AACBRB )=An¢=§
Problem 2. Suppose two dice are tossed and the numbers on the upper faces are observed. Let S
denote the set of all possible pairs that can be observed. For example, let (2, 3) denote that a 2 was
observed on the first die and a 3 on the second.
1. Define the following subsets of S: A: The number on the second die is even. B: The sume of
the two numbers is even. C: At least one number in the pair is odd. Using equally likely rule
to calculate P (A), P (B) and P (C).
2. List the points in A,
¯C, A \B, A \ ¯B,
¯A [B and
¯A \ C.
2
SI { ( ij ) : ii. 2.345.6, jil ,
2. 3. 4. 5.6 }
l' PC A) =L
.
PC B) = 3×36+1×4=21
PC c) = 1- PC both are even )=tY¥=¥
2. A={ ( i ,
2k ) : i. I. 2. 3. 4. 5.6,
k=l ,2,3 }
-
f- { ( i,
i ) : iii. 2.3.4516 # uB={ ( i. zjtl ) : 'tt :b
C 3 ,i ) : i=l ,
2. 3. 4. 5,6foil .2
,
C2i ,2jJ . i. I. 2.3
( 5,
i ) :it ,
2. 3. 4.516, II. 2.3 )
( 2 ,i ) : is 1. 3.5
( 4. i ) :i=l ,
3.5 At Atf @ zjtpiiti :b
" ' Emily;#IEEE;;;D eat
AAB = { cziizj ) : E- 1.2 ,} ,jH .
2.3)
An5= { C Ziti ,zj ) :ital '3j⇒2 , } }
Problem 3. Suppose two balanced coins are tossed and the upper faces are observed.
1. List the sample points for this experiment
2. Assign a reasonable probability to each sample point. (Are the sample points equally likely?)
3. Let A denote the event that exactly one head is observed and B the event that at least one
head is observed. List the sample points in both A and B.
4. From your answer to part 3., find P (A), P (B), P (A \B), P (A [B) and P (
¯A [B).
3
1. { ( HHI , CHT )
,CTHI ,
CTTHt t t t
2. ¥ 4¥ ¥
3. A={ C H .T ) ,
CTHI )
Be { CHHI,
CHTICTHI )
4.
PCAK 's,
PCBKYY
PLAABKK.
PCAVBKYY
PLEVB )=|