5 Confidence interval for the mean 89, 6 Actual confidence by simulation 90. 7 Testing problems first example 92, 8 Testing for the mean of a normal population 94. 9 Testing for the difference between means of two normal populations 95. 10 Testing for the mean in absence of normality 97. 11 Chi squared test for goodness of fit 98, 12 Tests for independence 100. 13 Regression and Linear regression 102, Appendix A Lecture by lecture plan 110. Appendix B Various pieces 111, Probability, 1 W HAT IS STATISTICS AND WHAT IS PROBABILITY. Sometimes statistics is described as the art or science of decision making in the face of uncertainty. Here are some examples to illustrate what it means. Example 1 Recall the apocryphal story of two women who go to King Solomon with a child each. claiming that it is her own daughter The solution according to the story uses human psychology. and is not relevant to recall here But is this a reasonable question that the king can decide. Daughters resemble mothers to varying degrees and one cannot be absolutely sure of guessing. correctly On the other hand by comparing various features of the child with those of the two. women there is certainly a decent chance to guess correctly. If we could always get the right answer or if we could never get it right the question would not. have been interesting However here we have uncertainty but there is a decent chance of getting. the right answer That makes it interesting for example we can have a debate between eyeists. and nosists as to whether it is better to compare the eyes or the noses in arriving at a decision. Example 2 The IISc cricket team meets the Basavanagudi cricket club for a match Unfortunately. the Basavanagudi team forgot to bring a coin to toss The IISc captain helpfully offers his coin but. can he be trusted What if he spent the previous night doctoring the coin so that it falls on one. side with probability 3 4 or some other number, Instead of cricket they could spend their time on the more interesting question of checking if. the coin is fair or biased Here is one way If the coin is fair in a large number of tosses common. sense suggests that we should get about equal number of heads and tails So they toss the coin. 100 times If the number of heads is exactly 50 perhaps they will agree that it is fair If the number. of heads is 90 perhaps they will agree that it is biased What if the number of heads is 60 Or 35. Where and on what basis to draw the line between fair and biased Again we are faced with the. question of making decision in the face of uncertainty. Example 3 A psychic claims to have divine visions unavailable to most of us You are assigned. the task of testing her claims You take a standard deck of cards shuffle it well and keep it face. down on the table The psychic writes down the list of cards in some order whatever her vision. tells her about how the deck is ordered Then you count the number of correct guesses If the. number is 1 or 2 perhaps you can dismiss her claims If it is 45 perhaps you ought to be take her. seriously Again where to draw the line, The logic is this Roughly one may say that surprise is just the name for our reaction to an event. that we a priori thought had low probability Thus we approach the experiment with the belief that. the psychic is just guessing at random and if the results are such that under that random guess. hypothesis they have very small probability then we are willing to discard our preconception and. accept that she is a psychic, How low a probability is surprising In the context of psychics let us say 1 10000 Once we fix. that we must find a number m 52 such that by pure guessing the probability to get more than. m correct guesses is less that 1 10000 Then we tell the psychic that if she gets more than m correct. guesses we accept her claim and otherwise reject her claim This raises the simple and you can. do it yourself, Question 4 For a deck of 52 cards find the number m such that. P by random guessing we get more than m correct guesses. Summary There are many situations in real life where one is required to make decisions under. uncertainty A general template for the answer could be to fix a small number that we allow. as the probability of error and deduce thresholds based on it This brings us to the question of. computing probabilities in various situations, Probability Probability theory is a branch of pure mathematics and forms the theoretical basis. of statistics In itself probability theory has some basic objects and their relations like real num. bers addition etc for analysis and it makes no pretense of saying anything about the real world. Axioms are given and theorems are then deduced about these objects just as in any other part of. mathematics, But a very important aspect of probability is that it is applicable In other words there are many. situations in which it is reasonable to take a model in probability. In the example above to compute the probability one must make the assumption that the deck. of cards was completely shuffled In other words all possible 52 orders of the 52 cards are. assumed to be equally likely Whether this assumption is reasonable or not depends on how well. the card was shuffled whether the psychic was able to get a peek at the cards whether some. insider is informing the psychic of the cards etc All these are non mathematical questions and. must be decided on other basis, However Probability and statistics are very relevant in many situations that do not involve any. uncertainty on the face of it Here are some examples. Example 5 Compression of data Large files in a computer can be compressed to a zip format. and uncompressed when necessary How is it possible to compress data like this To give a very. simple analogy consider a long English word like invertebrate If we take a novel and replace every. occurrence of this word with zqz then it is certainly possible to recover the original novel since. zqz does not occur anywhere else But the reduction in size by replacing the 12 letter word. by the 3 letter word is not much since the word invertebrate does not occur often Instead if we. replace the 4 letter word then by zqz then the total reduction obtained may be much higher. as the word then occurs quite often, This suggests the following optimal way to represent words in English The 26 most frequent. words will be represented by single letters The next 26 26 most frequent words will be repre. sented by two letter words the next 26 26 26 most frequent words by three letter words etc. Assuming there are no errors in transcription this is a good way to reduce the size of any text. document Now this involves knowing what the frequencies of occurrences of various words in. actual texts are Such statistics of usage of words are therefore clearly relevant and they could be. different for biology textbooks as compared to 19th century novels. Example 6 Search algorithms such as Google use many randomized procedures This cannot. be explained right now but let us give a simple reason to say why introducing randomness is a. good idea in many situations In the game of rock paper scissors two people simultaneously shout. one of the three words rock paper or scissors The rule is that scissors beats paper paper beats. rock and rock beats scissors if they both call the same word they must repeat In a game like. this although there is complete symmetry in the three items it would be silly to have a fixed. strategy In other words if you decide to always say rock thinking that it doesn t matter which. you choose then your opponent can use that knowledge to always choose paper and thus win. In many games where the opponent gets to know your strategy but not your move the best. strategy would involve randomly choosing your move. 2 D ISCRETE PROBABILITY SPACES, Definition 7 Let be a finite or countable1 set Let p 0 1 be a function such that. p 1 Then p is called a discrete probability space is called the sample space and. p are called elementary probabilities, Any subset A is called an event For an event A we define its probability as P A. Any function X R is called a random variable For a random variable we define its. expected value or mean as E X X p, All of probability in one line Take an interesting probability space p and an interesting. event A Find P A, This is the mathematical side of the picture It is easy to make up any number of probability. spaces simply take a finite set and assign non negative numbers to each element of the set so that. the total is 1, Example 8 0 1 and p0 p1 21 There are only four events here 0 1 and 0 1. Their probabilities are 0 1 2 1 2 and 1 respectively. Example 9 0 1 Fix a number 0 p 1 and let p1 p and p0 1 p The sample space is. the same as before but the probability space is different for each value of p Again there are only. four events and their probabilities are P 0 P 0 1 p P 1 p and P 0 1 1. 1For those unfamiliar with countable sets it will be explained in some detail later. Example 10 Fix a positive integer n Let, 0 1 n 1 n with i 0 or 1 for each i n. Let p 2 n for each Since has 2n elements it follows that this is a valid assignment of. elementary probabilities, There are 2 22 events One example is Ak and 1 n k where k is. some fixed integer In words Ak consists of those n tuples of zeros and ones that have a total of k. many ones Since there are nk ways to choose where to place these ones we see that Ak nk. Consequently, X Ak n 2 n if 0 k n, 2 0 otherwise, It will be convenient to adopt the notation that b 0 if a b are positive integers and if b a or. 2 n without having to split the values of k into, if b 0 Then we can simply write P Ak k. Example 11 Fix two positive integers r and m Let, 1 r with 1 i m for each i r. The cardinality of is mr since each co ordinate i can take one of m values Hence if we set. p m r for each we get a valid probability space, Of course there are 2m many events which is quite large even for small numbers like m 3. and r 4 Some interesting events are A r 1 B i 6 1 for all i C. i 6 j if i 6 j The reason why these are interesting will be explained later Because of. equal elementary probabilities the probability of an event S is just S mr. Counting A We have m choices for each of 1 r 1 There is only one choice for r. Hence A mr 1 Thus P A mr m, Counting B We have m 1 choices for each i since i cannot be 1 Hence B m 1 r. and thus P B mr 1 m, Counting C We must choose a distinct value for each 1 r This is impossible if. m r If m r then 1 can be chosen as any of m values After 1 is chosen there are. m 1 possible values for 2 and then m 2 values for 3 etc all the way till r which. has m r 1 choices Thus C m m 1 m r 1 Note that we get the same. answer if we choose i in a different order it would be strange if we did not. m m 1 m r 1, Thus P C mr Note that this formula is also valid for m r since one. of the factors on the right side is zero, 2 1 Probability in the real world In real life there are often situations where there are several. possible outcomes but which one will occur is unpredictable in some way For example when we. toss a coin we may get heads or tails In such cases we use words such as probability or chance. event or happening randomness etc What is the relationship between the intuitive and mathematical. meanings of words such as probability or chance, In a given physical situation we choose one out of all possible probability spaces that we think. captures best the chance happenings in the situation The chosen probability space is then called a. model or a probability model for the given situation Once the model has been chosen calculation of. probabilities of events therein is a mathematical problem Whether the model really captures the. given situation or whether the model is inadequate and over simplified is a non mathematical. question Nevertheless that is an important question and can be answered by observing the real. life situation and comparing the outcomes with predictions made using the model2. Now we describe several random experiments a non mathematical term to indicate a real. life phenomenon that is supposed to involve chance happenings in which the previously given. examples of probability spaces arise Describing the probability space is the first step in any prob. ability problem, Example 12 Physical situation Toss a coin Randomness enters because we believe that the coin. may turn up head or tail and that it is inherently unpredictable. The corresponding probability model Since there are two outcomes the sample space 0 1. where we use 1 for heads and 2 for tails is a clear choice What about elementary probabilities. Under the equal chance hypothesis we may take p0 p1 12 Then we have a probability model. for the coin toss, If the coin was not fair we would change the model by keeping 0 1 as before but letting. p1 p and p0 1 p where the parameter p 0 1 is fixed. Which model is correct If the coin looks very symmetrical then the two sides are equally likely. to turn up so the first model where p1 p0 2 is reasonable However if the coin looks irregular. then theoretical considerations are usually inadequate to arrive at the value of p Experimenting. with the coin by tossing it a large number of times is the only way. There is always an approximation in going from the real world to a mathematical model For. as the probability of error and deduce thresholds based on it This brings us to the question of computing probabilities in various situations Probability Probability theory is a branch of pure mathematics and forms the theoretical basis of statistics In itself probability theory has some basic objects and their relations like real num

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