# 7.2 Problems on distribution and density functions

(See Exercises 3 and 4 from "Problems on Random Variables and Probabilities"). The class $\left\{{C}_{j}:1\le j\le 10\right\}$ is a partition. Random variable X has values $\left\{1,3,2,3,4,2,1,3,5,2\right\}$ on C 1 through C 10 , respectively, with probabilities 0.08, 0.13, 0.06, 0.09, 0.14, 0.11, 0.12, 0.07, 0.11, 0.09.Determine and plot the distribution function F X .

T = [1 3 2 3 4 2 1 3 5 2];pc = 0.01*[8 13 6 9 14 11 12 7 11 9];[X,PX] = csort(T,pc);ddbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot

(See Exercise 6 from "Problems on Random Variables and Probabilities"). A store has eight items for sale. The prices are $3.50,$5.00, $3.50,$7.50, $5.00,$5.00, $3.50, and$7.50, respectively.A customer comes in. She purchases one of the items with probabilities 0.10, 0.15, 0.15, 0.20, 0.10 0.05, 0.10 0.15. Therandom variable expressing the amount of her purchase may be written

$X=3.5{I}_{{C}_{1}}+5.0{I}_{{C}_{2}}+3.5{I}_{{C}_{3}}+7.5{I}_{{C}_{4}}+5.0{I}_{{C}_{5}}+5.0{I}_{{C}_{6}}+3.5{I}_{{C}_{7}}+7.5{I}_{{C}_{8}}$

Determine and plot the distribution function for X .

T = [3.5 5 3.5 7.5 5 5 3.5 7.5];pc = 0.01*[10 15 15 20 10 5 10 15];[X,PX] = csort(T,pc);ddbn Enter row matrix of VALUES XEnter row matrix of PROBABILITIES PX % See MATLAB plot

(See Exercise 12 from "Problems on Random Variables and Probabilities"). The class $\left\{A,\phantom{\rule{0.166667em}{0ex}}B,\phantom{\rule{0.166667em}{0ex}}C,\phantom{\rule{0.166667em}{0ex}}D\right\}$ has minterm probabilities

$pm=0.001*\left[5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}7\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}6\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}8\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}9\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}14\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}22\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}33\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}21\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}32\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}50\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}75\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}86\phantom{\rule{0.277778em}{0ex}}129\phantom{\rule{0.277778em}{0ex}}201\phantom{\rule{0.277778em}{0ex}}302\right]$

Determine and plot the distribution function for the random variable $X={I}_{A}+{I}_{B}+{I}_{C}+{I}_{D}$ , which counts the number of the events which occur on a trial.

 npr06_12 Minterm probabilities in pm, coefficients in c T = sum(mintable(4)); % Alternate solution. See Exercise 12 from "Problems on Random Variables and Probabilities" [X,PX]= csort(T,pm); ddbnEnter row matrix of VALUES X Enter row matrix of PROBABILITIES PX % See MATLAB plot

Suppose a is a ten digit number. A wheel turns up the digits 0 through 9 with equal probability on each spin. On ten spins what is the probabilityof matching, in order, k or more of the ten digits in a , $0\le k\le 10$ ? Assume the initial digit may be zero.

$P=\mathtt{c}\mathtt{b}\mathtt{i}\mathtt{n}\mathtt{o}\mathtt{m}\left(10,0.1,0:10\right)$ .

In a thunderstorm in a national park there are 127 lightning strikes. Experience shows that the probability of of a lightning strike starting a fire is about0.0083. What is the probability that k fires are started, $k=0,1,2,3$ ?

P = ibinom(127,0.0083,0:3) P = 0.3470 0.3688 0.1945 0.0678

A manufacturing plant has 350 special lamps on its production lines. On any day, each lamp could fail with probability $p=0.0017$ . These lamps are critical, and must be replaced as quickly as possible. It takes about one hour toreplace a lamp, once it has failed. What is the probability that on any day the loss of production time due to lamp failaures is k or fewer hours, $k=0,\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}3,\phantom{\rule{0.277778em}{0ex}}4,\phantom{\rule{0.277778em}{0ex}}5\text{?}$

P = 1 - cbinom(350,0.0017,1:6)

= 0.5513 0.8799 0.9775 0.9968 0.9996 1.0000

Two hundred persons buy tickets for a drawing. Each ticket has probability 0.008 of winning. What is the probability of k or fewer winners, $k=2,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}4\text{?}$

P = 1 - cbinom(200,0.008,3:5) = 0.7838 0.9220 0.9768

Two coins are flipped twenty times. What is the probability the results match (both heads or both tails) k times, $0\le k\le 20$ ?

P = ibinom(20,1/2,0:20)

#### Questions & Answers

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
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Damian Reply
absolutely yes
Daniel
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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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket 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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