Intro to Probabiltiy theory notes for Elements Class.pptx

If a random sample of 25 people is drawn from this

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blind. If a random sample of 25 people is drawn from this population. Find the probability that: a) Five or fewer will be color blind P(X≤ 5) =.9666 b) Six or more will be color blind The probability that six or more are color blind is the complement of the probability that five or fewer are not color blind. P( X ≥6)=1- P(X ≤ 5)=1-.9666=.0334 c) Between six and nine inclusive will be color blind P(6 ≤ X ≤ 9)=P(X ≤ 9)-P(X ≤ 5)=.9999-.9666=.0333 d) Two, three or four will be color blind P(2 ≤ X ≤ 4)=P(X ≤) – P(X ≤ 1)=.9020-.2712=.6308
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68 The Poisson Distribution Under certain conditions the Binomial Distribution assumes a very convenient form. These conditions are: 1. Sample size is large n ≥ 50 2. p = Probability of event occurring is very small p ≤ 0.1 3. The mean or average λ is a finite number λ = n p is ≤ 5 When these conditions are satisfied, it can be shown mathematically that the binomial reduces to a convenient form, where Pr(x) the probability of x successes or events happening is given by: x! λ e Pr(x) x λ Where λ is the mean or average value of the distribution. This expression is known as the Poisson Law
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69 The Poisson Distribution This distribution has been used extensively as a probability model in biology and medicine. If x is the number of occurrences of some random events in an interval of time or space, the probability that x will occur is: x=0,1,2……, λ>0 λ = parameter (average number of occurrence of the random event in the interval e= constant =2.7183 f(x) = e λ x X !
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70 Characteristics defining a Poisson random variable Poisson distribution occurs when there are events which don’t occur as outcome of a definite numbers of trials of an experiment but which occur at random point of time and space and our interest lies in the occurrence of the event only. The experiment consists of counting the number x of times a particular event occurs during a given unit of time The probability that an event occurs in a given unit of time is the same for all units. The number of events that occur in one unit of time is independent of the number that occur in other units of time.
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71 Theoretically an infinite number of occurrence of the event must be possible in the interval. The Poisson event is an infrequently occurring events with very small probability of occurrence. Some situations where the Poisson process can be employed successfully are No. of deaths from a disease such as heart attack or cancer or due to snake bite No. of suicides reported in a particular city. No. of Bank failures in a given year No. of lightning strikes in a certain area.
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72 Example: In a study of suicide it was found that monthly distribution of suicide in Cook country, between 1977 and 1987 closely followed a poisson distribution with parameter λ= 2.75. Find the probability that a randomly selected month will be one in which three adolescent suicide occurs.
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