Probability, Binomial, Poisson, Normal, and Sampling Distributions
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| Elemental Properties of Probability | P(Ei)>0; all events must have a probability greater than or equal to 0 of occurring. Can't be higher than 1.
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| Elemental Properties of Probability | The sum of the probabilities of all mutually exclusive outcomes is equal to 1. P(E1)+{(E2)+...+p(En)= 1
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| Elemental Properties of Probability | For any two mutually exclusive events A and B, the probability of the occurrence of A or B is equal to the sum of their individual probabilities. P(A or B) = P(A) + P(B). Use union symbol.
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| Marginal Probability | Probability of one event happening.
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| Joint Probability | The probability that a subject picked at random possesses two characteristics at the same time. Use intersect symbol.
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| Addition Rule | P(A or B) = P(A) + P(B) - P(A and B). use union symbol.
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| Unconditional Probability | Refers to a probability that includes the total group. The denominator is the total group.
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| Conditional Probability | Only involves a subset of a total group. The denominator is only a subset of the group.P(A|B)=P(A and B)/P(B) and P(B|A)=P(A and B)/P(A)
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| Multiplication Rule | Probability that is computed from other probabilities. P(A and B) = P(B) x P(A|B)P(A and B) = P(A) x P(B|A)
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| Independent Events | Event A has occurred but has no effect on the probability of B. Probability is the same regardless of whether or not A occurs. Has to have non-zero probabilities.P(B|A)=P(B) and P(A|B)=P(A).
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| Multiplication Rule for Independent Events | If 2 events are independent, the probability of their joint occurrence is equal to the product of the probabilities of their single occurrences.P(A and B)=P(A)xP(B)
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| To test independence | P(A and B)=P(A)xP(B)P(A|B)=P(A)P(B|A)=P(B)
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| Complementary Events | The probability of event A is equal to 1 minus the probability of its complement, not A.P(A)=1-P(not A)P(not A)=1-P(A)
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| Binomial Distribution | Discrete probability distribution. Derived from a process known as Bernoulli Process (made up of a series of Bernoulli trials). P(X=x)=f(x)=nCx x p^x x q ^(n-x). Parameters:n and p
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| Bernoulli Process | Each Bernoulli trial results in one of two possible, mutually exclusive, outcomes. One of the possible outcomes is denoted as a success (p) and the other is denoted a failure (q). Experiment consists of n identical trials.
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| Bernoulli Process cont. | The probability of success remains constant from trial to trial. q=1-p. The trials are independent. The binomial random variable (x) is the count of the # of successes in the n trials.
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| Poisson Distribution | Discrete probability distribution usually associated with rare events. Derived from Poisson Process. Parameters: lambda
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| Poisson Process | 1. The occurrence of events are independent2. An infinite number of event occurrences should be possible in the time interval3. The probability of an event occurring in a certain time interval is proportional to the length of the time interval.
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| Normal Distribution | In general, when the number of values, n, approaches infinity and the width of the class intervals approaches zero, the frequency polygon becomes a smooth curve (normal). Parameters: mu and sigma.
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| Continuous Probability Distributions | The total area bounded by its curve and the x-axis is equal to 1 and the sub-area under the curve bounded by the curve, the x-axis,and the perpendiculars erected at any 2 points a and b gives the probability that X is between the two points a and b.
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| Characteristics of the Normal Distribution | 1. Symmetrical about its mean2. Mean, median, and mode are all equal3. The total area under the curve is one square unit4. + or - 1 sd is 68% of total area+ o - 2 sd is 95% of total area+ or - 3 sd is 99.7% of total area
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| Standard Normal Distribution | mean=0standard deviation=1standard normal random variable=zZ transformation can transform any normal random variable into a standard normal distribution. The z-score transforms a data value into the # of standard deviations that value is from mean.
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| Sampling Distributions | 1. Allows us to answer probability questions about sample statistics.2. Provide the necessary theory for making statistical inference procedures valid.Histogram and frequency polygon form a peak.
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| Sampling Distribution of a Statistic | The distribution of all possible values than can be assumed by some statistic, computed from samples of the same size, randomly drawn from the same population.
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| Constructing a Sampling Distribution | 1. Choose population, same size, and statistic2. Draw a simple random sample with replacement3.Compute the statistic for the sample4. Repeat steps 2 and 35. Take the statistics and construct the relative frequency distribution for the statistic
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| Sampling from a Normally Distributed Population | 1. The distribution of x bar will be normal.2. The mean of the sampling dist. will be equal to the mean of the population3. The variance of the sampling dist. will be equal to the variance of the pop. divided by the sample size.
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| Sampling from a Non-Normally Distributed Population | Central Limit Theorem: Given a pop. of any non-normal form, the sampling distribution of x bar will be approximately normally distributd when the sample size is large. Works when n is greater than or equal to 30.
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| Distribution of the Sampling Proportion | population proportion:psample proportion: p-hat1. When sample size is large, dist. of p-hat is approx. normally dist.2. The mean of the dist. is equal to the true pop. proportion p3. The variance of samp. dist. of p-hat: p(1-p)/n np>5, n(1-p)>5
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