Intro State Notes Week Five
Intro State Notes Week Five TMATH 110 C
University of Washington Tacoma
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This 0 page Class Notes was uploaded by Qihua Wu on Sunday November 1, 2015. The Class Notes belongs to TMATH 110 C at University of Washington Tacoma taught by KENNEDY,MAUREEN C. in Fall 2015. Since its upload, it has received 14 views. For similar materials see Intro Stat Applications in Math at University of Washington Tacoma.
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Date Created: 11/01/15
Probability our estimate of uncertainty the higher the probability the more likely an event is going to happen Event consists of one or more results of some processes Notation to represent the likelihood of event A happening is PA the probability of A Summation of all probabilities should be 1 Simple event event that has the simplest parts meaning there is only one way to get to the result Sample space include all possible distinct outcomes the number of sample space is equivalent to the summation of all simple events Probability principle When we say the probability of an event we are saying the probability between 0 and 1 inclusive If it is 0 then it is not happening If it is 1 it is certainly happening You can not have a negative probability or a probability greater than 1 If it is less than or equal to 005 it is unlikely to happen You can think of this with a normal distribution graph we say within 2 standard deviations are likely and 95 of the data falls within 2 standard deviations which means the excluded 5 are unlikely When we say an event is unlikely it means it has small probability When we say an event is unusual it means it is an extreme event Classical Approach Theoretical approach If the number of possible outcomes of an event is known and each unique outcome has an equal chance of happening then we can calculate the probability of an individual event PA number of ways A can happen number of possible outcomes Approximating probabilities using relative frequencies Empirical approach When the unique outcomes do not have equal chance of appearing we can do an experiment and compare the recorded frequency of an event with the number of tests in the experiment Round the result to 3 decimals A P number of times A happened number of times a test was repeated quot means rounding Subjective probability based on the experiences on the event estimate the probability of that event Difference between the relative frequency and subjective probability is that relative frequency is based on data while subjective is based on experiences Complementary Events For any event any other event that is exclusive of the event is called its complement For complementary events the sum is always 1 because it includes entire events the event itself and all the other events that do not include the event Law of large numbers the state that as the number of trials increases the mean value of the experiment is closer to the true mean value which makes the impact of unusual events less important in determining the mean If we repeat the experiment of sample size the results would be similar but not identical unless we are conducting experiment of the entire population Misconception of law of large numbers The Gambler39s Fallacy law of large number means that the impact of the streaks gets wash out as the number of trials increases not that a streak is an indication of the next possibility in fact there is no relation between the previous outcome and the next outcome Random Variable X Variable that is randomly chosen each time the trial is repeated there is a possibility of different outcome can be discrete or continuous Discrete Random Variable nite number Continuous Random Variable In nitely many values Fixed Variable Same value even if the trial is repeated Realization of a random process x the event either occurred or not after a random process is done We then compare the possible outcome and the actual outcome to nd the frequency the possible outcome would happen Probability distribution Px can be de ne as either nite or in nite random variables and gives the probability for each outcome of the random variable X Each probability is between 0 and 1 inclusive All probabilities add up to 1 Can nd mean variance and standard deviation by square rooting the va anceL Mean is the expected value meaning after in nitely many experiments the expected average value should be the mean Continuous probability distribution The probability an outcome takes a certain value is 0 The total area under the curve adds up to 1 Every point on the curve has to be greater than or equal to 0 Which is pretty much similar to a nite probability distribution where all possibilities have to add up to 1 and no possibilities below 0 however because this is continuous it could have in nitely many values so it would be impossible to get a certain value When we determine the possibilities of continuous random variables we are determining the probability between a range of values instead ofjust an individual value since the probability an outcome takes a certain value is 0 Uniform distribution continuous random variable where the probability of the values are evenly distributed across the range of possible values ab meaning within the possible range of data every interval of a given length has the same probability To nd the probability that is not uniform we nd the area under the curve the density Normal distribution bellshaped curve I12A D Dquot22A2 Where l is mean and ljquot2 is variance Similar to uniform distribution normal distribution can only calculate probabilities associated with intervals and not individual values by calculating the area of the interval under the curve does not crosses the xaxis since probability can not be less than 0 Properties of Normal Distribution 1 Bell shaped symmetric 2 Completely speci ed by the mean the height and the variance the spread 3 Centered at the mean 4 The height of the curve at X mean the highest curve is determined by the standard deviation 5 The distance between the mean and the in ection point is the standard deviation the in ection points are mean plus standard deviation and mean minus standard deviation 6 The limits of the distribution are in nities both positive and negative despite that in the real world often time the limit is 0 Simpli ed Notation XN l jquot2 can be read as quotX is normally distributed between the mean and the va riancequot