This preview shows pages 1–12. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Stat400 Lecture @131" 12nd,, 2011 Page 1 Practice problems 1. Heidi and Alex have agreed to meet for lunch between
noon (0:00 pm.) and 1:00 pm. Denote Heidi’s ar—
rival time by X, Alex’s by Y, and suppose X and Y
are independent with probability density functions 3$20§$§1 _ 5y40§y§1
fX($> _ l 0 otherwise fY<y) _ l 0 otherwise (a) Write doyvn the joint p.d.f. of (X, Y). I, / ondepwdwt >0, '
JEN/w : fx(xlfl/(<(l> : “1:74, Ogre/pew (b) Find the probability that Heidi arrives before Alex. W X < X l 3 5 7c yyyy @an ( felt: . O Man—ram V . ..V_s.,_ Stat400 Lecture 22, Apr 12nd, 2011 Page 3 3. Suppose that commercial airplane crashes in a certain
country occur at the rate of 2.5 per year. Assume
that such crashes follow a Poisson process, (a) Starting from now, let X denotes the time un—
til the next crash, what is the distribution of X?
Write down the p.d.f. of X. rate. A .: 2‘ g Cmf/iesnu I
Xm Iii/NW)? : 7V "04 yEW/mxﬁ'
fol”) C blgé Q“ /. an (b) What is the probabilitythat the next crash will
occ u r wit h i n 5th.,reewmont,hsl) Brno/47M; :1 U'Mjéd/l/ ﬂair.
Wm 02$») :: F(0,2§~):/Q ﬁg
0‘25; Stat400 Lecture 22, Apr 12nd,, 2011 Page 2 2. According to an airline industry report, roughly ,1
piece of luggage out of every 200 that are checked is
lost. Suppose that a frequent—flying businesswoman
will be checking 120 bags over the course of the next
year. Using Poisson approximation to find the prob—
ability that she will lose 2 or more pieces of luggage. X f 4:6 luggage am: We? /20 ﬁgs x” [9020/
Rx») : l~l>(X<l) 3:5) *9: Parka/l <05) Q l a [fr 0/ /» iggmm‘mzaﬂ Porgy)” Fm gag 3 / Wan wave/We #aW/zﬂ » a
~ . rrrrr w»! ’ I I (9pm
I MILK/n M ' “ * I
X 1% 6 ” 00/7/15 W1€(§/JQCQ)/ X/v R9159”! M w marge #MW%€T M27739 mm to Oak/me Mt amt (/A/EXFa?) 0g (3? WUMa magi??? 750 (9526/08 (Xi/L WK Stat400 Lecture 22, Apr 12nd7 2011 Page «1 4.(a) The tensile strength X of paper has ,u :55; and a =A§‘(pounglsmper square inch). A random sam—
ple of sizeGi = 10ms taken from the distribution
. W I u u
of tensrle strengths.w Compute the probability that
the sample mean is greater than 29.4 pounds
per square inch. (Hint: use Central limit theorem) Stat400 Lecture 22., Apr 12nd; 2011 Page 5 (b) Let X equal the number out of n : 48 mature
aster seeds that will germinate when p = 0.75 is the probability that a particular seed germinates.
Approximate P(35 g X g 40) using normal dis— tribution. (MW/f, rim—pr
><m M45? 075“) RJ/A/(sa 7)
:nprrja)
Wgysx :40) : Flegtof pwqjj
(/ecfm / 8 ) r , Stat400 Lecture 22, Apr 12nd; 2011 Page 6 5. This is farmer Bates’s first year in the pumpkin busi—
ness, and his pumpkins have a range of sizes, with
diameters distributed normally with mean 131 and
standard deviation Farmer Boggis has been in
the business for three years: his; _ ve di—
ameters that are N(14, in inches. ' 'L/lwlﬁpé/Mlﬁ‘r/C * (a) What is the fraction of Bates's pumpkins t at are
above 15” in diameter? alga/,1 5W yf: [gotta/g
l>(><>if> X ” NM” 22/ WW; * (b) A pumpkin less than d inches in diameter is not
saleable‘ This year 5% of Bates’s pumpkins were
not saleable. What is the value of d? “‘1 P< Mel) : 00f
I X43 0&3)
"3s 2 Stat400 Lecture 22,_Ap1“ 12nd; 201.1 Page 7 (c) Let X be the size of a pumpkin chosen at random
from Bates' lot, and let MAR a pumpkin chosen at
random from Boggis's lot. What is the probability
, I that Bates’s umpkin is larger than Bo gis's? x~Mi %, 22), ym/v’i / k w r/omepwgm Xr/m/VGA 2o) P<><>/) awn/>0) Stat400 Lecture 22‘. Apr 12nd? 2011 Page 8 6. Let X1, ’2, ’3, X4 be a random sample of size n 2 4
from the distribution with probability density function fxiivi : “Ema 55 E (a) Find P{maX(X1,)g;i—)‘(3) > 1) :1: [Km P(nia>('(Xz/Xz/<32 S} > Stat400 Lecture 22, Apr 12nd} 2011 Page 9 43’
(b) A sample range is defined as R : meme21727374 Xi— miniZ127374 Xi, find the expected value of R, Le. {H E[R].
. I ‘ :: my Xz/Xl/X 2 mMHIMMXc < 2— ] K.)
W XC :m (Mme x3, 2%)
LII/Z/ka ,
Em :— 3 ~ Hm, 2a) ...
View
Full
Document
This note was uploaded on 09/11/2011 for the course STAT 400 taught by Professor Tba during the Spring '05 term at University of Illinois at Urbana–Champaign.
 Spring '05
 TBA
 Statistics, Probability

Click to edit the document details