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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) ...
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 Statistics, Normal Distribution, Probability, Probability theory, probability density function, Bates, Stat400 Lecture

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