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Suppose X is a nonnegative, absolutely continuous random variable. Let Z = g ( X ) = C e - a X , where a > 0 , C > 0 . Then 0 < Z C . Use properties of the exponential and natural log function to show that

F Z ( v ) = 1 - F X - ln ( v / C ) a for 0 < v C

Z = C e - a X v iff e - a X v / C iff - a X ln ( v / C ) iff X - ln ( v / C ) / a , so that

F Z ( v ) = P ( Z v ) = P ( X - ln ( v / C ) / a ) = 1 - F X - ln ( v / C ) a
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Use the result of [link] to show that if X exponential ( λ ) , then

F Z ( v ) = v C λ / a 0 < v C
F Z ( v ) = 1 - 1 - e x p - λ a · ln ( v / C ) = v C λ / a
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Present value of future costs. Suppose money may be invested at an annual rate a , compounded continually. Then one dollar in hand now, has a value e a x at the end of x years. Hence, one dollar spent x years in the future has a present value e - a x . Suppose a device put into operation has time to failure (in years) X exponential ( λ ) . If the cost of replacement at failure is C dollars, then the present value of the replacement is Z = C e - a X . Suppose λ = 1 / 10 , a = 0 . 07 , and C = $ 1000 .

  1. Use the result of [link] to determine the probability Z 700 , 500 , 200 .
  2. Use a discrete approximation for the exponential density to approximate the probabilities in part (a). Truncate X at 1000 and use 10,000 approximation points.
P ( Z v ) = v 1000 10 / 7
v = [700 500 200];P = (v/1000).^(10/7) P = 0.6008 0.3715 0.1003tappr Enter matrix [a b]of x-range endpoints [0 1000] Enter number of x approximation points 10000Enter density as a function of t 0.1*exp(-t/10) Use row matrices X and PX as in the simple caseG = 1000*exp(-0.07*t); PM1 = (G<=700)*PX' PM1 = 0.6005PM2 = (G<=500)*PX' PM2 = 0.3716PM3 = (G<=200)*PX' PM3 = 0.1003
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Optimal stocking of merchandise. A merchant is planning for the Christmas season. He intends to stock m units of a certain item at a cost of c per unit. Experience indicates demand can be represented by a random variable D Poisson ( μ ) . If units remain in stock at the end of the season, they may be returned with recovery of r per unit. If demand exceeds the number originally ordered, extra units may be ordered at a cost of s each. Units are sold at a price p per unit. If Z = g ( D ) is the gain from the sales, then

  • For t m , g ( t ) = ( p - c ) t - ( c - r ) ( m - t ) = ( p - r ) t + ( r - c ) m
  • For t > m , g ( t ) = ( p - c ) m + ( t - m ) ( p - s ) = ( p - s ) t + ( s - c ) m

Let M = ( - , m ] . Then

g ( t ) = I M ( t ) [ ( p - r ) t + ( r - c ) m ] + I M ( t ) [ ( p - s ) t + ( s - c ) m ]
= ( p - s ) t + ( s - c ) m + I M ( t ) ( s - r ) ( t - m )

Suppose μ = 50 m = 50 c = 30 p = 50 r = 20 s = 40. .
Approximate the Poisson random variable D by truncating at 100. Determine P ( 500 Z 1100 ) .

mu = 50; D = 0:100;c = 30; p = 50;r = 20; s = 40;m = 50; PD = ipoisson(mu,D);G = (p - s)*D + (s - c)*m +(s - r)*(D - m).*(D<= m); M = (500<=G)&(G<=1100); PM = M*PD'PM = 0.9209[Z,PZ] = csort(G,PD); % Alternate: use dbn for Zm = (500<=Z)&(Z<=1100); pm = m*PZ'pm = 0.9209
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(See Example 2 from "Functions of a Random Variable") The cultural committee of a student organization has arranged a special deal for tickets to a concert. The agreement is that the organization will purchase tentickets at $20 each (regardless of the number of individual buyers). Additional tickets are available according to the following schedule:

  • 11-20, $18 each
  • 21-30, $16 each
  • 31-50, $15 each
  • 51-100, $13 each

If the number of purchasers is a random variable X , the total cost (in dollars) is a random quantity Z = g ( X ) described by

g ( X ) = 200 + 18 I M 1 ( X ) ( X - 10 ) + ( 16 - 18 ) I M 2 ( X ) ( X - 20 ) +
( 15 - 16 ) I M 3 ( X ) ( X - 30 ) + ( 13 - 15 ) I M 4 ( X ) ( X - 50 )
where M 1 = [ 10 , ) , M 2 = [ 20 , ) , M 3 = [ 30 , ) , M 4 = [ 50 , )

Suppose X Poisson (75). Approximate the Poisson distribution by truncating at 150. Determine P ( Z 1000 ) , P ( Z 1300 ) , and P ( 900 Z 1400 ) .

X = 0:150; PX = ipoisson(75,X);G = 200 + 18*(X - 10).*(X>=10) + (16 - 18)*(X - 20).*(X>=20) + ... (15 - 16)*(X- 30).*(X>=30) + (13 - 15)*(X - 50).*(X>=50); P1 = (G>=1000)*PX' P1 = 0.9288P2 = (G>=1300)*PX' P2 = 0.1142P3 = ((900<=G)&(G<=1400))*PX' P3 = 0.9742[Z,PZ] = csort(G,PX); % Alternate: use dbn for Zp1 = (Z>=1000)*PZ' p1 = 0.9288
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Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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