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This module provides a number of homework exercises related to Discrete Random Variables.

1. Complete the PDF and answer the questions.

x size 12{x} {} P ( X = x ) size 12{P \( X=x \) } {} x P ( X = x ) size 12{x cdot P \( X=x \) } {}
0 0.3
1 0.2
2
3 0.4

  • Find the probability that X = 2 size 12{X=2} {} .
  • Find the expected value.

  • 0.1
  • 1.6

Suppose that you are offered the following “deal.” You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6.

  • What are you ultimately interested in here (the value of the roll or the money you win)?
  • In words, define the Random Variable X size 12{X} {} .
  • List the values that X size 12{X} {} may take on.
  • Construct a PDF.
  • Over the long run of playing this game, what are your expected average winnings per game?
  • Based on numerical values, should you take the deal? Explain your decision in complete sentences.

A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars.

  • Construct a PDF for each investment.
  • Find the expected value for each investment.
  • Which is the safest investment? Why do you think so?
  • Which is the riskiest investment? Why do you think so?
  • Which investment has the highest expected return, on average?
  • $200,000;$600,000;$400,000
  • third investment
  • first investment
  • second investment

A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase 4 tickets. The prize is 2 passes to a Broadway show, worth a total of $150.

  • What are you interested in here?
  • In words, define the Random Variable X size 12{X} {} .
  • List the values that X size 12{X} {} may take on.
  • Construct a PDF.
  • If this fund-raiser is repeated often and you always purchase 4 tickets, what would be your expected average winnings per game?

Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X size 12{X} {} = the number of children

x size 12{x} {} P ( X = x ) size 12{P \( X=x \) } {} x P ( X = x ) size 12{x cdot P \( X=x \) } {}
0 0.10
1 0.20
2 0.30
3
4 0.10
5 0.05
6 (or more) 0.05

  • Find the probability that a married adult has 3 children.
  • In words, what does the expected value in this example represent?
  • Find the expected value.
  • Is it more likely that a married adult will have 2 – 3 children or 4 – 6 children? How do you know?
  • 0.2
  • 2.35
  • 2-3 children

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given below.

x size 12{x} {} P ( X = x ) size 12{P \( X=x \) } {}
3 0.05
4 0.40
5 0.30
6 0.15
7 0.10

  • In words, define the Random Variable X size 12{X} {} .
  • What does it mean that the values 0, 1, and 2 are not included for X size 12{X} {} on the PDF?
  • On average, how many years do you expect it to take for an individual to earn a B.S.?

Questions & Answers

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
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
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
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Source:  OpenStax, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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