|-.
Midterm #3
Mathcmatics 5OI0. I, Summer 2006
Departmert of Mathematics, Unive$ity o{ Utah
Juiy 5, 2006
+ This is a.lc€d book, clced-rote qanindtion+ This exm beejnsat 1:00p.D. and endsar 2:00p.m. shep.
+ ThG *m is nad€ ur of 4 qnmtions for a total of 40 points.
* Write you dses
clearly. lf you shov me.ely a nunerical answer' then you are likely to rcceire zero
p a d i a l . r e d i l . S o s l o w a n d e x p l d i ny . , , , q o , 1 , .
* Conffne your work to tbis wo.kshetYou nay nse boLh sjdes of tle paper. Thde is tlso an exha sh@t of
papei for you to w.jte o, if 'ru wish, dd a normal table at ihe lery back.
1. {10 pointstotal) Slpposethat tbe height,in inches,of a 25-year-oldman is a.normal random ria.ble
dirh meanlr 1t nid rdtianceo" 625.
(a) (5 points) what perceDtaseof 2s-year old meD are over 6 feet 2 jnches tall?
= a- t
?Ns.>,t'tJ
l- Qt{.?) '-
t- o'88+{
=rni[il
t--'-
(b) (5 poinis) W}at percentage of men in the 6-footer club are over &foot 5 iDches?
?F"'t
?ltrnl
-'
lx''lL\
?Ft'z"l
= l-O@'..{) s 1-o'qrr
1-+Q'q)
1 - 0 . 65 s {
I
ta+-+t \
=
r-+ t-,=-d
t " - -
'_+s#1
: '
2. (10 points total) For al (l,6 > 0 define
ato ,,t-
lo',"-'tt
,to'a'
(a) (5 potfts) Prove that if a, D > 0 are fixed p:'.meters, then rhe foltowins is a probabitity density
function:
r@)'=lEC:d*
r ( 1 x ) b - lj r o < ' < r '
othe.wise.
[0
- -{-(.-l 2. o
-
fl
+t-r tx- -
|
'fo'
"i-'cr-'')*t
3(4b)
=^ '
dz<-
(b) (5 points)Conputetlxl.
E{s.l-- l] -'
{
'.^-'(^-*f-'e*
B(4'b)
f.( g*n-' (l-*l-',r,.
G(a,b)
--@)
v93--/
2
3. (10 points) r X is unifoffnly distributedon (0, 1) then what is the densityfunction of y :: ex ?
( a )= ? [ f
\.
s^] =
- C9 - a
1
{
0
n^"]
"[rs
030^a St
l^^>t
P-* <o
tw\r1
+
g *<0*-e-3 t
fr("1
0
(<a S€ -,
i+ ^>o
a
o
l----1---------{
z
Lt
L
7
h*-L
Z = f-^"
"t-S.rt u",*f - [orl1 .
7 r-o +|d
4. (10 points) There is a stick oflength, vhich ve may think of. mathematically,as the inteNal [O,r].
Choosea point at rardom along the stick (e.g., let the point be unifomly disttbuted in the inte al
[0 , ,]), and brea] the siick at that point. Let x denote the ratio of the lensth of the lonser piece to
the shorterone. Find P{x < t/4}.
rf z st
rc-^ -S=*
t
,1 z>t"- {'^
, i,'1 <.) 22. +L
4*t
Z=}_
L-x
No,rr i{
? ( r <? \ = ,
L <t
O^:O* 'lt\A |*od , i{
1L{"-
*
'l{
<L*_-' z<
,t
L>l
1A.^ - + L
1+ t_
."0'"r,
ff-1?l:-1
L2'
\.? L
b9-ct+-
>t
/).t.
d^,t (: .t.
++L