Differences and similarities_of_discrete_probability_distributions

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Differences and Differences and SimilaritiesSimilarities

Discrete Random VariablesDiscrete Random Variables

Some Problems tooSome Problems too

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Probability DistributionsProbability Distributions

A probability distribution is a statement of A probability distribution is a statement of a probability function that assigns all the a probability function that assigns all the probabilities associated with a random probabilities associated with a random variable. variable. – A discrete probability distribution is a A discrete probability distribution is a

distribution of discrete random variables (that distribution of discrete random variables (that is, random variables with a limited set of is, random variables with a limited set of values). values).

– A continuous probability distribution is A continuous probability distribution is concerned with a random variable having an concerned with a random variable having an infinite set of values.infinite set of values.

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BinomialBinomial

The experiment must have only two The experiment must have only two possible outcomes (success & failure)possible outcomes (success & failure)

The probability of success must be The probability of success must be constant from trial to trial (each member constant from trial to trial (each member of the sample have same of the sample have same pp))

Independence must be maintained (no Independence must be maintained (no trial’s outcome influences another)trial’s outcome influences another)

WeWe countcount the number ofthe number of successessuccesses in in nn trialstrials

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Explain why the following are Explain why the following are not Binomial experimentsnot Binomial experiments

I draw 3 cards from an ordinary deck and count I draw 3 cards from an ordinary deck and count X, the number of aces. Drawing is done without X, the number of aces. Drawing is done without replacement.replacement.

A couple decides to have children until a girl is A couple decides to have children until a girl is born. Let X denote the number of children the born. Let X denote the number of children the couple will have.couple will have.

In a sample of 5000 individuals, I record the age In a sample of 5000 individuals, I record the age of each person, denoted as X .of each person, denoted as X .

A chemist repeats a solubility test ten times on A chemist repeats a solubility test ten times on the same substance. Each test is conducted at the same substance. Each test is conducted at a temperature 10 degrees higher than the a temperature 10 degrees higher than the previous test. Let X denote the number of times previous test. Let X denote the number of times the substance dissolves completely.the substance dissolves completely.

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BinomialBinomial In a small clinical trial with 20 patients, let In a small clinical trial with 20 patients, let

X denote the number of patients that X denote the number of patients that respond to a new skin rash treatment. The respond to a new skin rash treatment. The physicians assume independence among physicians assume independence among the patients. Here, X ~ bin (n = 20; p), the patients. Here, X ~ bin (n = 20; p), where p denotes the probability of where p denotes the probability of response to the treatment. In a statistics response to the treatment. In a statistics problem, p might be an unknown problem, p might be an unknown parameter that we might want to parameter that we might want to estimate. For this problem, we'll assume estimate. For this problem, we'll assume that p = 0.7. We want to compute (a). P(X that 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]

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PoissonPoisson

If Binomial If Binomial pp is small and is small and nn is large, is large, Poisson can be used as an approximation Poisson 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 interval in an interval of time or space or volume. [Only mean is of 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 service Number of customers arriving at a service

centre in a certain periodcentre in a certain period

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PoissonPoisson It is useful for describing It is useful for describing

– radioactive decay (number of particles emitted radioactive decay (number of particles emitted in a fixed period of time);in a fixed period of time);

– the number of vacancies in the Supreme Court the number of vacancies in the Supreme Court each year;each year;

– the numbers of dye molecules taken up by the 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 facility the number of customers arriving at a facility – The number of earthquakes in a certain area The number of earthquakes in a certain area

per yearper year

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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]

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GeometricGeometric

Under the same conditions of the Under the same conditions of the Binomial, the Geometric counts the Binomial, 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 before Probability you take the course 3 times before

you pass (x = 3)you pass (x = 3)– Probability the police will stop 10 cars before Probability 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 find Probability I screen 5 applicants before I find

the first qualified (x = 5) the first qualified (x = 5)

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MultinomialMultinomial

Similar to the Binomial except that:Similar to the Binomial except that: The experiment will have more than The experiment will have more than

two possible outcomes (Xtwo possible outcomes (X11, X, X22, …, X, …, Xnn)) The probability of each outcome will The probability of each outcome will

be given (pbe given (p11, p, p22, …, p, …, pnn).). The sample will cover all the The sample will cover all the

outcomesoutcomes

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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?

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Fidelity sells a small SUV called the Fidelity sells a small SUV called the Nissan X-Trail. They believe that Nissan X-Trail. They believe that they have 20% of the small SUV they have 20% of the small SUV market. Assume it is true. What is market. Assume it is true. What is the probability that in a random the probability that in a random sample of 15 small SUV owners, 5 sample 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?

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Fidelity sells a small SUV called the Fidelity sells a small SUV called the Nissan X-Trail. They believe that Nissan X-Trail. They believe that they have 20% of the small SUV they have 20% of the small SUV market. Assume it is true. What is market. Assume it is true. What is the probability that they would have the 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?

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Identify the DistributionIdentify the Distribution Assume that the small SUV market is Assume that the small SUV market is

divided as shown in the table. What is the divided as shown in the table. What is the probability that in a random sample of 40 probability that in a random sample of 40 small SUV’s at the toll booth, 8 were small SUV’s at the toll booth, 8 were Nissan; 10 were Honda; 9 were Toyota; 7 Nissan; 10 were Honda; 9 were Toyota; 7 were Suzuki and 6 were other?were Suzuki and 6 were other?

BrandNissan X-Trail

Honda CRV

Toyota Rav4

SuzukiVitara Other

% 0.12 0.27 0.22 0.25 0.14

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Insurance companies keep track of Insurance companies keep track of accidents as part of their risk accidents as part of their risk management. Suppose that lady management. Suppose that lady drivers have a 2 percent chance of drivers have a 2 percent chance of committing an accident in the year. committing an accident in the year. A random sample of 1,000 ladies was A random sample of 1,000 ladies was examined – what is the probability examined – what is the probability that 10 of these ladies committed an that 10 of these ladies committed an accident?accident?

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A tailor was contracted to make suits A tailor was contracted to make suits for a wedding party. He discovered for a wedding party. He discovered that the material chosen had athat the material chosen had a reputationreputation of having 3 defects per of having 3 defects per square metre. square metre. – What is the probability that in a square What 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 square square

metres, 20 defects were found?metres, 20 defects were found?

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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 viewing Channel 12 has 50 percent of the viewing audience, Channel 10 has 30 percent of audience, Channel 10 has 30 percent of the viewing audience, and Channel 3 has the viewing audience, and Channel 3 has 20 percent of the viewing audience. Find 20 percent of the viewing audience. Find the probability that among eight television the probability that among eight television views in that city, randomly chosen on a views in that city, randomly chosen on a Saturday night, five will be watching Saturday night, five will be watching Channel 12, two will be watching Channel Channel 12, two will be watching Channel 10, and one will be watching Channel 310, and one will be watching Channel 3

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My car has a dead battery and I need My car has a dead battery and I need some jumper cables. People stop to help some jumper cables. People stop to help me but I have to refuse their help if they me but I have to refuse their help if they have no jumper cables. Suppose 10% of have no jumper cables. Suppose 10% of the people driving on the road have the people driving on the road have jumper cables.jumper cables.– What is the probability that the first person What is the probability that the first person

who can help me is the 8who can help me is the 8thth person who person who stopped?stopped?

– How many persons do you expect to stop How 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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Differences and similarities_of_discrete_probability_distributions