Differences and similarities_of_discrete_probability_distributions

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Differences and similarities_of_discrete_probability_distributions

  1. 1. 11 Differences andDifferences and SimilaritiesSimilarities Discrete Random VariablesDiscrete Random Variables Some Problems tooSome Problems too
  2. 2. 22 Probability DistributionsProbability Distributions  A probability distribution is a statement ofA probability distribution is a statement of a probability function that assigns all thea probability function that assigns all the probabilities associated with a randomprobabilities associated with a random variable.variable. – A discrete probability distribution is aA discrete probability distribution is a distribution of discrete random variables (thatdistribution of discrete random variables (that is, random variables with a limited set ofis, random variables with a limited set of values).values). – A continuous probability distribution isA continuous probability distribution is concerned with a random variable having anconcerned with a random variable having an infinite set of values.infinite set of values.
  3. 3. 33 BinomialBinomial  The experiment must have only twoThe experiment must have only two possible outcomes (success & failure)possible outcomes (success & failure)  The probability of success must beThe probability of success must be constant from trial to trial (each memberconstant from trial to trial (each member of the sample have sameof the sample have same pp))  Independence must be maintained (noIndependence must be maintained (no trial’s outcome influences another)trial’s outcome influences another)  WeWe countcount the number ofthe number of successessuccesses inin nn trialstrials
  4. 4. 44 Explain why the following areExplain why the following are not Binomial experimentsnot Binomial experiments  I draw 3 cards from an ordinary deck and countI draw 3 cards from an ordinary deck and count X, the number of aces. Drawing is done withoutX, the number of aces. Drawing is done without replacement.replacement.  A couple decides to have children until a girl isA couple decides to have children until a girl is born. Let X denote the number of children theborn. Let X denote the number of children the couple will have.couple will have.  In a sample of 5000 individuals, I record theIn a sample of 5000 individuals, I record the age of each person, denoted as X .age of each person, denoted as X .  A chemist repeats a solubility test ten times onA chemist repeats a solubility test ten times on the same substance. Each test is conducted atthe same substance. Each test is conducted at a temperature 10 degrees higher than thea temperature 10 degrees higher than the previous test. Let X denote the number of timesprevious test. Let X denote the number of times the substance dissolves completely.the substance dissolves completely.
  5. 5. 55 BinomialBinomial  In a small clinical trial with 20 patients, letIn a small clinical trial with 20 patients, let X denote the number of patients thatX denote the number of patients that respond to a new skin rash treatment. Therespond to a new skin rash treatment. The physicians assume independence amongphysicians assume independence among the patients. Here, X ~ bin (n = 20; p),the patients. Here, X ~ bin (n = 20; p), where p denotes the probability ofwhere p denotes the probability of response to the treatment. In a statisticsresponse to the treatment. In a statistics problem, p might be an unknownproblem, p might be an unknown parameter that we might want toparameter that we might want to estimate. For this problem, we'll assumeestimate. For this problem, we'll assume that p = 0.7. We want to compute (a). P(Xthat p = 0.7. We want to compute (a). P(X = 15), (b) P(X≥ 15), and (c) P(X < 10).= 15), (b) P(X≥ 15), and (c) P(X < 10). [0.1789; 0.416; 0.017][0.1789; 0.416; 0.017]
  6. 6. 66 PoissonPoisson  If BinomialIf Binomial pp is small andis small and nn is large,is large, Poisson can be used as an approximationPoisson can be used as an approximation ((pp ≤ 0.05 and≤ 0.05 and nn ≥ 20;≥ 20; μμ ==nn**pp))  OtherwiseOtherwise, by itself, the Poisson, by itself, the Poisson countscounts the number ofthe number of occurrencesoccurrences in an intervalin an interval of time or space or volume. [Only mean isof time or space or volume. [Only mean is given]. Examplegiven]. Example – Number of accidents in a dayNumber of accidents in a day – No. of tears (defects) in a sq metre of clothNo. of tears (defects) in a sq metre of cloth – Number of customers arriving at a serviceNumber of customers arriving at a service centre in a certain periodcentre in a certain period
  7. 7. 77 PoissonPoisson  It is useful for describingIt is useful for describing – radioactive decay (number of particles emittedradioactive decay (number of particles emitted in a fixed period of time);in a fixed period of time); – the number of vacancies in the Supreme Courtthe number of vacancies in the Supreme Court each year;each year; – the numbers of dye molecules taken up bythe numbers of dye molecules taken up by small particles;small particles; – the sizes of colloidal particles;the sizes of colloidal particles; – the number of accidents per unit timethe number of accidents per unit time – the number of customers arriving at a facilitythe number of customers arriving at a facility – The number of earthquakes in a certain areaThe number of earthquakes in a certain area per yearper year
  8. 8. 88 PoissonPoisson  Phone calls arrive at a switchboard according to a Poisson process, at a rate of = 3 per minute. – Find the probability that 8 or fewer calls come in during a 5-minute span. – What is the average number of calls in a 5-minute span? – [0.037; 15]
  9. 9. 99 GeometricGeometric  Under the same conditions of theUnder the same conditions of the Binomial, the Geometric counts theBinomial, the Geometric counts the number of failuresnumber of failures beforebefore (until) the(until) the firstfirst successsuccess – hence there is no sample size.– hence there is no sample size. – Probability you take the course 3 times beforeProbability you take the course 3 times before you pass (x = 3)you pass (x = 3) – Probability the police will stop 10 cars beforeProbability the police will stop 10 cars before they find the suspect (x = 10)they find the suspect (x = 10) – Probability I screen 5 applicants before I findProbability I screen 5 applicants before I find the first qualified (x = 5)the first qualified (x = 5)
  10. 10. 1010 MultinomialMultinomial  Similar to the Binomial except that:Similar to the Binomial except that:  The experiment will have more thanThe experiment will have more than two possible outcomes (Xtwo possible outcomes (X11, X, X22, …, X, …, Xnn))  The probability of each outcome willThe probability of each outcome will be given (pbe given (p11, p, p22, …, p, …, pnn).).  The sample will cover all theThe sample will cover all the outcomesoutcomes
  11. 11. 1111 Identify the DistributionIdentify the Distribution  In the following examples:In the following examples: – Identify the distributionIdentify the distribution – Find the probabilityFind the probability – What are the expected values?What are the expected values?
  12. 12. 1212 Identify the DistributionIdentify the Distribution  Fidelity sells a small SUV called theFidelity sells a small SUV called the Nissan X-Trail. They believe thatNissan X-Trail. They believe that they have 20% of the small SUVthey have 20% of the small SUV market. Assume it is true. What ismarket. Assume it is true. What is the probability that in a randomthe probability that in a random sample of 15 small SUV owners, 5sample of 15 small SUV owners, 5 are X-Trails?are X-Trails?  How many do you expect to find?How many do you expect to find?
  13. 13. 1313 Identify the DistributionIdentify the Distribution  Fidelity sells a small SUV called theFidelity sells a small SUV called the Nissan X-Trail. They believe thatNissan X-Trail. They believe that they have 20% of the small SUVthey have 20% of the small SUV market. Assume it is true. What ismarket. Assume it is true. What is the probability that they would havethe probability that they would have to interview 6 small SUV owners,to interview 6 small SUV owners, (randomly selected) before they find(randomly selected) before they find the first X-Trail owner?the first X-Trail owner?  How many do you expect to find?How many do you expect to find?
  14. 14. 1414 Identify the DistributionIdentify the Distribution  Assume that the small SUV market isAssume that the small SUV market is divided as shown in the table. What is thedivided as shown in the table. What is the probability that in a random sample of 40probability that in a random sample of 40 small SUV’s at the toll booth, 8 weresmall SUV’s at the toll booth, 8 were Nissan; 10 were Honda; 9 were Toyota; 7Nissan; 10 were Honda; 9 were Toyota; 7 were Suzuki and 6 were other?were Suzuki and 6 were other? Brand Nissan X-Trail Honda CRV Toyota Rav4 Suzuki Vitara Other % 0.12 0.27 0.22 0.25 0.14
  15. 15. 1515 Identify the DistributionIdentify the Distribution  Insurance companies keep track ofInsurance companies keep track of accidents as part of their riskaccidents as part of their risk management. Suppose that ladymanagement. Suppose that lady drivers have a 2 percent chance ofdrivers have a 2 percent chance of committing an accident in the year.committing an accident in the year. A random sample of 1,000 ladiesA random sample of 1,000 ladies was examined – what is thewas examined – what is the probability that 10 of these ladiesprobability that 10 of these ladies committed an accident?committed an accident?
  16. 16. 1616 Identify the DistributionIdentify the Distribution  A tailor was contracted to make suitsA tailor was contracted to make suits for a wedding party. He discoveredfor a wedding party. He discovered that the material chosen had athat the material chosen had a reputationreputation of having 3 defects perof having 3 defects per square metre.square metre. – What is the probability that in a squareWhat is the probability that in a square metre examined, 4 defects were seen?metre examined, 4 defects were seen? – What is the probability that inWhat is the probability that in 1010 squaresquare metres, 20 defects were found?metres, 20 defects were found?
  17. 17. 1717 Identify the DistributionIdentify the Distribution  A certain city has three television stations.A certain city has three television stations. During prime time on Saturday nights,During prime time on Saturday nights, Channel 12 has 50 percent of the viewingChannel 12 has 50 percent of the viewing audience, Channel 10 has 30 percent ofaudience, Channel 10 has 30 percent of the viewing audience, and Channel 3 hasthe viewing audience, and Channel 3 has 20 percent of the viewing audience. Find20 percent of the viewing audience. Find the probability that among eight televisionthe probability that among eight television views in that city, randomly chosen on aviews in that city, randomly chosen on a Saturday night, five will be watchingSaturday night, five will be watching Channel 12, two will be watching ChannelChannel 12, two will be watching Channel 10, and one will be watching Channel 310, and one will be watching Channel 3
  18. 18. 1818 Identify the DistributionIdentify the Distribution  My car has a dead battery and I needMy car has a dead battery and I need some jumper cables. People stop to helpsome jumper cables. People stop to help me but I have to refuse their help if theyme but I have to refuse their help if they have no jumper cables. Suppose 10% ofhave no jumper cables. Suppose 10% of the people driving on the road havethe people driving on the road have jumper cables.jumper cables. – What is the probability that the first personWhat is the probability that the first person who can help me is the 8who can help me is the 8thth person whoperson who stopped?stopped? – How many persons do you expect to stopHow many persons do you expect to stop before I can find one who is able to help?before I can find one who is able to help?

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