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Customers arrive at a service center with independent interarrival times in hours, which have exponential (3) distribution. The time X for the third arrival is thus gamma ( 3 , 3 ) . Without using tables or m-programs, determine P ( X 2 ) .

P ( X 2 ) = P ( Y 3 ) , Y poisson ( 3 · 2 = 6 )
P ( Y 3 ) = 1 - P ( Y 2 ) = 1 - e - 6 ( 1 + 6 + 36 / 2 ) = 0 . 9380
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Five people wait to use a telephone, currently in use by a sixth person. Suppose time for the six calls (in minutes) are iid, exponential (1/3).What is the distribution for the total time Z from the present for the six calls? Use an appropriate Poisson distribution to determine P ( Z 20 ) .

Z gamma (6,1/3).

P ( Z 20 ) = P ( Y 6 ) , Y poisson ( 1 / 3 · 20 )
P ( Y 6 ) = c p o i s s o n ( 20 / 3 , 6 ) = 0 . 6547
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A random number generator produces a sequence of numbers between 0 and 1. Each of these can be considered an observed value of a random variableuniformly distributed on the interval [0, 1]. They assume their values independently.A sequence of 35 numbers is generated. What is the probability 25 or more are less than or equal to 0.71? (Assume continuity. Do not make a discrete adjustment.)

p = cbinom(35,0.71,25) = 0.5620

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Five “identical” electronic devices are installed at one time. The units fail independently, and the time to failure, in days, of each is a randomvariable exponential (1/30). A maintenance check is made each fifteen days. What is the probability that at least four are still operating at the maintenance check?

p = exp(-15/30) = 0.6065 P = cbinom(5,p,4) = 0.3483

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Suppose X N ( 4 , 81 ) . That is, X has gaussian distribution with mean μ = 4 and variance σ 2 = 81 .

  1. Use a table of standardized normal distribution to determine P ( 2 < X < 8 ) and P ( | X - 4 | 5 ) .
  2. Calculate the probabilities in part (a) with the m-function gaussian.
  1. P ( 2 < X < 8 ) = Φ ( ( 8 - 4 ) / 9 ) - Φ ( ( 2 - 4 ) / 9 ) =
    Φ ( 4 / 9 ) + Φ ( 2 / 9 ) - 1 = 0 . 6712 + 0 . 5875 - 1 = 0 . 2587
    P ( | X - 4 | 5 ) = 2 Φ ( 5 / 9 ) - 1 = 1 . 4212 - 1 = 0 . 4212
  2. P1 = gaussian(4,81,8) - gaussian(4,81,2) P1 = 0.2596P2 = gaussian(4,81,9) - gaussian(4,84,-1) P2 = 0.4181
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Suppose X N ( 5 , 81 ) . That is, X has gaussian distribution with μ = 5 and σ 2 = 81 . Use a table of standardized normal distribution to determine P ( 3 < X < 9 ) and P ( | X - 5 | 5 ) . Check your results using the m-function gaussian.

P ( 3 < X < 9 ) = Φ ( ( 9 - 5 ) / 9 ) - Φ ( ( 3 - 5 ) / 9 ) = Φ ( 4 / 9 ) + Φ ( 2 / 9 ) - 1 = 0 . 6712 + 0 . 5875 - 1 = 0 . 2587
P ( | X - 5 | 5 ) = 2 Φ ( 5 / 9 ) - 1 = 1 . 4212 - 1 = 0 . 4212
P1 = gaussian(5,81,9) - gaussian(5,81,3) P1 = 0.2596P2 = gaussian(5,81,10) - gaussian(5,84,0) P2 = 0.4181
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Suppose X N ( 3 , 64 ) . That is, X has gaussian distribution with μ = 3 and σ 2 = 64 . Use a table of standardized normal distribution to determine P ( 1 < X < 9 ) and P ( | X - 3 | 4 ) . Check your results with the m-function gaussian.

P ( 1 < X < 9 ) = Φ ( ( 9 - 3 ) / 8 ) - Φ ( ( 1 - 3 ) / 9 ) =
Φ ( 0 . 75 ) + Φ ( 0 . 25 ) - 1 = 0 . 7734 + 0 . 5987 - 1 = 0 . 3721
P ( | X - 3 | 4 ) = 2 Φ ( 4 / 8 ) - 1 = 1 . 3829 - 1 = 0 . 3829
P1 = gaussian(3,64,9) - gaussian(3,64,1) P1 = 0.3721P2 = gaussian(3,64,7) - gaussian(3,64,-1) P2 = 0.3829
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Items coming off an assembly line have a critical dimension which is represented by a random variable N(10, 0.01). Ten items are selected at random. What is the probability that three or more are within 0.05 of themean value μ .

p = gaussian(10,0.01,10.05) - gaussian(10,0.01,9.95) p = 0.3829P = cbinom(10,p,3) P = 0.8036
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The result of extensive quality control sampling shows that a certain model of digital watches coming off a production line have accuracy, in seconds per month,that is normally distributed with μ = 5 and σ 2 = 300 . To achieve a top grade, a watch must have an accuracy within the range of -5 to +10 secondsper month. What is the probability a watch taken from the production line to be tested will achieve top grade? Calculate, using a standardized normal table. Checkwith the m-function gaussian.

P ( - 5 < X < 10 ) = Φ ( 5 / 300 ) + Φ ( 10 / 300 ) - 1 = Φ ( 0 . 289 ) + Φ ( 0 . 577 ) - 1 = 0 . 614 + 0 . 717 - 1 = 0 . 331

P = g a u s s i a n ( 5 , 300 , 10 ) - g a u s s i a n ( 5 , 300 , - 5 ) = 0 . 3317
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Use the m-procedure bincomp with various values of n from 10 to 500 and p from 0.01 to 0.7, to observe the approximation of the binomial distribution by the Poisson.

Experiment with the m-procedure bincomp.

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Use the m-procedure poissapp to compare the Poisson and gaussian distributions. Use various values of μ from 10 to 500.

Experiment with the m-procedure poissapp.

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Random variable X has density f X ( t ) = 3 2 t 2 , - 1 t 1 (and zero elsewhere).

  1. Determine P ( - 0 . 5 X < 0 , 8 ) , P ( | X | > 0 . 5 ) , P ( | X - 0 . 25 | 0 . 5 ) .
  2. Determine an expression for the distribution function.
  3. Use the m-procedures tappr and cdbn to plot an approximation to the distribution function.
3 2 t 2 = t 3 / 2
  1. P 1 = 0 . 5 * ( 0 . 8 3 - ( - 0 . 5 ) 3 ) = 0 . 3185 P 2 = 2 0 . 5 1 3 2 t 2 = ( 1 - ( - 0 . 5 ) 3 ) = 7 / 8
    P 3 = P ( | X - 0 . 25 | 0 . 5 ) = P ( - 0 . 25 X 0 . 75 ) = 1 2 [ ( 3 / 4 ) 3 - ( - 1 / 4 ) 3 ] = 7 / 32
  2. F X ( t ) = - 1 t f X = 1 2 ( t 3 + 1 )
  3. tappr Enter matrix [a b]of x-range endpoints [-1 1] Enter number of x approximation points 200Enter density as a function of t 1.5*t.^2 Use row matrices X and PX as in the simple casecdbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot
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Random variable X has density function f X ( t ) = t - 3 8 t 2 , 0 t 2 (and zero elsewhere).

  1. Determine P ( X 0 . 5 ) , P ( 0 . 5 X < 1 . 5 ) , P ( | X - 1 | < 1 / 4 ) .
  2. Determine an expression for the distribution function.
  3. Use the m-procedures tappr and cdbn to plot an approximation to the distribution function.
( t - 3 8 t 2 ) = t 2 2 - t 3 8
  1. P 1 = 0 . 5 2 / 2 - 0 . 5 3 / 8 = 7 / 64 P 2 = 1 . 5 2 / 2 - 1 . 5 3 / 8 - 7 / 64 = 19 / 32 P 3 = 79 / 256
  2. F X ( t ) = t 2 2 - t 3 8 , 0 t 2
  3. tappr Enter matrix [a b]of x-range endpoints [0 2] Enter number of x approximation points 200Enter density as a function of t t - (3/8)*t.^2 Use row matrices X and PX as in the simple casecdbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot
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Random variable X has density function

f X ( t ) = ( 6 / 5 ) t 2 for 0 t 1 ( 6 / 5 ) ( 2 - t ) for 1 < t 2 = I [ 0 , 1 ] ( t ) 6 5 t 2 + I ( 1 , 2 ] ( t ) 6 5 ( 2 - t )
  1. Determine P ( X 0 . 5 ) , P ( 0 . 5 X < 1 . 5 ) , P ( | X - 1 | < 1 / 4 ) .
  2. Determine an expression for the distribution function.
  3. Use the m-procedures tappr and cdbn to plot an approximation to the distribution function.
  1. P 1 = 6 5 0 1 / 2 t 2 = 1 / 20 P 2 = 6 5 1 / 2 1 t 2 + 6 5 1 3 / 2 ( 2 - t ) = 4 / 5
    P 3 = 6 5 3 / 4 1 t 2 + 6 5 1 5 / 4 ( 2 - t ) = 79 / 160
  2. F X ( t ) = 0 t f X = I [ 0 , 1 ] ( t ) 2 5 t 3 + I ( 1 . 2 ] ( t ) [ - 7 5 + 6 5 ( 2 t - t 2 2 ) ]
  3. tappr Enter matrix [a b]of x-range endpoints [0 2] Enter number of x approximation points 400Enter density as a function of t (6/5)*(t<=1).*t.^2 + ... (6/5)*(t>1).*(2 - t) Use row matrices X and PX as in the simple casecdbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot
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Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

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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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