Chapter 5 Discrete Probability Distributions Binomial Poisson p x p n n px 1 p n x x n x x of success p probability of successes on trial n of trials f x probability of x success in n trials p x e x x expected value or mean of occurrence f x probability of x success in n trials 1 Sequence of n identical trials 2 2 possible outcomes for each trial success failure 3 The probability of success p does not change from trial to trial 4 The probability of failure 1 p does not change from trial to trial 1 Probability of occurrence is same for any 2 intervals of equal length 2 Occurrence nonoccurrence in any interval is independent of the occurrence in any other 5 Trials are independent not affected by others interval Given probability of successes sample size EXAMPLES OF DISCRETE RANDOM VARIABLES Given average per unit basis no sample size EXAMPLES OF CONTINUOUS RANDOM VARIABLES Random variable A numerical description of the outcome of an experiment Discrete random variable A random variable that may assume either a finite number of values or an infinite sequence of values Continuous random variable A random variable that may assume any numerical value in an interval or collection of intervals Probability distribution A description of how the probabilities are distributed over the values of the random variable Probability function A function denoted by f x that provides the probability that x assumes a particular value for a discrete random variable Discrete uniform probability distribution A probability distribution for which each possible value of the random variable has the same probability Expected value A measure of the central location of a random variable Variance A measure of the variability or dispersion of a random variable Standard deviation The positive square root of the variance Binomial experiment An experiment having the four properties stated at the beginning of Section 5 4 Binomial probability distribution A probability distribution showing the probability of x successes in n trials of a binomial experiment Binomial probability function The function used to compute binomial probabilities Poisson probability distribution A probability distribution showing the probability of x occurrences of an event over a specified interval of time or space Poisson probability function The function used to compute Poisson probabilities Chapter 6 Continuous Probability Distributions z x mean standard deviation z z score z x np np 1 p 1 Differentiated by 2 parameters the mean and the stand deviation 2 Highest point on the normal curve is the mean median mode 3 The mean of distribution can be any numerical value 0 4 3 normal distributions w same standard deviation can have 3 different means 5 Symmetric skewness 0 6 Larger values wider flatter curves showing more variability 7 Probabilities for normal random variable are given by areas under the normal curve Total area under curve 1 8 68 3 1 standard deviation 95 4 2 standard deviations 99 7 3 standard deviations Probability density function A function used to compute probabilities for a continuous random variable The area under the graph of a probability density function over an interval represents probability Uniform probability distribution A continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length Normal probability distribution A continuous probability distribution Its probability density function is bell shaped and determined by its mean and standard deviation Standard normal probability distribution A normal distribution with a mean of zero and a standard deviation of one Continuity correction factor A value of 5 that is added to or subtracted from a value of x when the continuous normal distribution is used to approximate the discrete binomial distribution Exponential probability distribution A continuous probability distribution that is useful in computing probabilities for the time it takes to complete a task Chapter 7 Sampling and Sampling distributions Mean z x n mean standard deviation n sample size z z score Proportion z p p p 1 p n p sample proportion x of elements in sample n sample size p mean population proportion Point estimate p x n Sampled population The population from which the sample is taken Frame A listing of the elements the sample will be selected from Parameter A numerical characteristic of a population such as a population mean a population standard deviation a population proportion p and so on Simple random sample A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected Random sample A random sample from an infinite population is a sample selected such that the following conditions are satisfied 1 Each element selected comes from the same population 2 each element is selected independently Sampling without replacement Once an element has been included in the sample it is removed from the population and cannot be selected a second time Sampling with replacement Once an element has been included in the sample it is returned to the population A previously selected element can be selected again and therefore may appear in the sample more than once Sample statistic A sample characteristic such as a sample mean a sample standard deviation s a sample proportion and so on The value of the sample statistic is used to estimate the value of the corresponding population parameter Stratified random sampling A probability sampling method in which the population is first divided into strata and a simple random sample is then taken from each stratum Point estimator The sample statistic such as s or that provides the point estimate of the population parameter Point estimate The value of a point estimator used in a particular instance as an estimate of a population parameter Target population The population for which statistical inference such as point estimates are made It is important for the target population to correspond as closely as possible to the sampled population Sampling distribution A probability distribution consisting of all possible values of a sample statistic Unbiased A property of a point estimator that is present when the expected value of the point estimator is equal to the population parameter it estimates Finite population correction factor The term that is used in the formulas