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Stat 330 (Spring 2015): slide set 1 2 Because statistical inference makes use of probability models, probability is a foundation for statistics. To use probability and statistics in a mathematically coherent way, we need a formal framework for talking about random processes and the elements that comprise random experiments. Statistics - theory of information that uses data to make inferences about questions of interest, under the assumption that there is a random component to the process that generated the data. Probability - mathematical theory for modeling experiments where outcomes occur randomly. We want to study physical processes that are not completely deterministic. Using probability and statistics to understand the random components of such processes can help us do this. Two deﬁnitions (among many that I’ve seen) of probability and statistics are as follows. Probability and Statistics Last update: January 9, 2015 Stat 330 (Spring 2015) Slide set 1 Some examples of what we consider to be random experiments are below. More interesting examples can be found in the textbook or Prof. Hofmann’s notes. • 3 The term experiment used here does not necessarily have its usual meaning of a controlled situations in which outputs (responses) are observed as a result of inputs (factors). A name for such a process is a random experiment. Many physical processes involve a random component – an element that cannot be described exactly by a deterministic algorithm. • • • Probabilistic Models for “Real-World” Processes Stat 330 (Spring 2015): slide set 1 1 Uncertainty appears in all areas of computer science and engineering. We will see some realistic examples as we proceed. Uncertainty exists in many aspects of science, business and our everyday life. It is something usually unavoidable which we have to deal with. • • The lack of certainty, a state of having limited knowledge where it is impossible to exactly describe current state or future outcome, (or the existence of more than one possible outcome). Stat 330 (Spring 2015): slide set 1 • Uncertainty/Randomness Stat 330 (Spring 2015): slide set 1 A company measures the installation time of a software system under diﬀerent conditions so that it can give customers some idea of the time required. • • Consumers can send an email to an organization’s phishing box to report a phishing attempt. The organization records the number of notiﬁcations and the time between notiﬁcations. • Components of Random Experiments (Continued) 6 Ω = {ordered triples of U’s and D’s} = {DDD, DDU, DU D, U DD, U U D, U DU, DU U, U U U } and |Ω| = 8 = 23 Ω = (0, ∞) 3. Network Routers: Ω = {H, T H, T T H, T T T H, . . .} 2. Time to access webpage: 1. Toss coin until a head: Examples: Sample Space (Ω) - set of all possible outcomes. ω = (router 1 down, router 2 down, recipient computer up) = DDU ω = 3.527 seconds 3. A message can take two network routers to get to a recipient computer. We may record the status of router 1, the status of router 2, and the status of the recipient computer, where the status is either up (U) or down (D). 7 Example 1. Toss coin until a head: Ω = {1, 2, 3, . . .} is an equivalent expression for the sample space. – Note that there are usually multiple ways to express the sample space for a particular experiment. 1. Toss coin until a head: Discrete 2. Time to access webpage: Not discrete 3. Network Routers: Discrete Examples: – Discrete Sample Space - sample space with a ﬁnite or countably inﬁnite number of elements. Stat 330 (Spring 2015): slide set 1 We try to access a web page and record the time it takes for the webpage to respond. • ω = TTTTH 2. Record the time for a webpage to respond. Stat 330 (Spring 2015): slide set 1 The Wall Street Journal tracks the DOW Jones industrial averages. • 1. Toss a coin until we get a head. Elementary Outcome (ω) - an outcome of a random process. Examples: 5 Record the number of car accidents at an intersection. • • Stat 330 (Spring 2015): slide set 1 Components of Random Experiments 4 Record the result of tossing two coins repeatedly: HH,HT,TH,TH,HH,... • Deﬁnition: A random experiment is a process with random outcomes. Examples of Random Experiments • Stat 330 (Spring 2015): slide set 1 C = Successful transmission = {DU U, U DU, U U U } 8 3. Suppose the message is transmitted successfully if at least one router is up and the recipient’s computer is up. B = More than 10 seconds = (10, ∞) 2. Time to access a webpage: A = ﬁrst head occurs between 5 and 11 tosses (inclusive) = {5, 6, 7, 8, 9, 10, 11} 1. Toss coin until a head: Event (A) - A ⊂ Ω i.e. subset of Ω. (A is a collection of elementary outcomes). Examples: Components of Probability Experiments (continued)