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1、1,Chapter 1 A preliminary reviewWhat is statistics?Statistics is the science and practice dealing with random phenomena. Statistics, based on statistical theory, is concerned with collecting and processing data, summarizing information, estimating descriptive constants (parameters), discovering empi
2、rical laws, testing hypotheses, and designing experiments in such a way that valid inferences can be drawn from empirical evidence.Within statistical theory, randomness and uncertainty are modeled by probability theory.,2,ExperimentLets consider tossing a coin. We will consider only two possible out
3、comes: either a head (H) appears, or a tail (T) appears.If a coin is tossed once, there are two possible outcomes: H or T. If a coin is tossed twice, there are 22 = 4 possible outcomes: HH, HT, TH, TT, where HT means a head occurs on the first toss and a tail on the second.Generalizing this exercise
4、, we may refer to the tossing of a coin (whether it is tossed once, twice, or n times) as one experiment. Since tossing a coin three times may be considered to be an experiment and is a composite of three separate experiments where a coin is tossed only once each time, we may refer to the shorter ex
5、periments as trials and the collection of the trials as “the experiment”.In general, an experiment is the process of following a well-defined set of rules, where the result of following those rules is not known prior to the experiment.Question: How many outcomes are there if a coin is tossed 3 times
6、?,3,ProbabilityDefinition 1: The sample space is the collection of all possible different outcomes of an experiment. Definition 2: A point in the sample space is a possible outcome of an experiment.Definition 3: An event is any set of points in the sample space.Questions:Write down the sample space
7、if an experiment consists of tossing a coin twice.If an examination consists of 10 “true or false” questions, how many points are there in the sample space?Refer to question 1, write down the events of “two heads”, “at least one head”, “no head”.,4,Probability (contd)To each point in the sample spac
8、e there corresponds a number called the probability of the point or the probability of the outcome. These probabilities is a number between 0 and 1.Definition: If A is an event associating with an experiment, and if nA represents the number of times A occurs in n independent repetitions of the exper
9、iment, the probability of the event A, denoted by p(A), is:Interpretation of probabilities:Frequency interpretation: the proportion of the number of times that head appears in tossing a coin (in the long run).Subjective (personal) belief: probability represents a degree of belief in a proposition, b
10、ased on all the information. It applies especially when there is little or no direct evidence, e.g., prior in Bayesian method.,5,Probability (contd)Exercises:In an experiment of tossing an unbiased coin once, write down the probability of p(H) and p(T). How to interpret these probabilities?Now lets
11、toss the coin three times, what are the following probabilities p(3 tails) =p(at least one head) =p(more heads than tails) = Find the probability to draw an ace from an ordinary deck of 52 playing cards.Find the probability to draw two aces from the same deck of the cards. (Note there are two ways t
12、o draw two aces: without replacement and with replacement.),6,Conditional probabilityIf A and B are two events in a sample space, the event “both A and B occur” is called the joint event A and B and is represented by AB. Here C = AB. Then the Probability of “A given B” is given by the probability of
13、 “AB” relative to the reduced sample space “B”, expressed as:Example: Consider rolling a die. Let S be the sample space, let A be the event “a 4, 5, or 6 occurs”, and let B the event “an even number (2, 4, 6) occurs”. Then the probability of A given B isQuestion: Toss a coin three times, then find t
14、he probability for the event “three heads”, given there are at least two heads.,7,Independent eventsThe concept of independence is closely related to conditional probability. We say A and B are independent, if . This formula can be easily derived from the conditional probabilities:This derivation is
15、 based on the independent property, , i.e., if the probability of A, given that B occurs, is the same as the probability of A, regardless of the occurrence or non-occurrence of B, then the occurrence of A is independent of B.Similarly, we have .Check if the following two events are independent: In a
16、n experiment of tossing a balanced coin twice, let A be the event “a head occurs on the first toss”, and let B be the event “a head occurs on the second toss”.,8,Random variablesA random variable is a function that assigns real numbers to the points in a sample space.Example: Either head or tail wil
17、l appear in tossing a coin. The sample space consists of two points representing these two possible outcomes. Let X (upper case) denote a random variable. We can assign 1 to X if head appears, but 0 to X if tail appears. If the coin is unbiased, we will have the probabilitiesAn important feature of a random variable: Before we toss the coin, we dont know if head or tail will appear.,
