1 EE Three + Two = Five

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1 Indian Institute of Technology Bombay Department of Electrical Engineering Handout 2 EE 325 Probability and Random Processes Cricket and Randomness July 21, EE Three + Two = Five Learning probability theory is laborious, from one end of the canal to the other where there is light, and then all you do is CRY!. When paced well, with less courses and more time, it is fun to learn. Now, 5:5: is a time you should be in the playing grounds, having a crack at the slushy footy goal-posts, before making your way to a mouth-awakening dinner, made by real cr/ooks. What you do not know is that your glorified chefs were perhaps carpenters last year, and gardeners the year before. IIT is good at recycling. Deprived of all these luxuries, instead you find yourselves bottled in this gigantic room, listening to some random instructor. Well, that is a good start! Randomness is the soul of this course, and better the instructor be a bit random. Lemme throw some random questions, which can gobble up your evening cricket and messes your dinner. Example 1 If you play cricket on random evenings (not the same as randomly play cricket every evening), what is the chance that you fail this course due to insufficient attendance? Solution: you may not like to know this, nevertheless EE325 will teach you how to compute this. Example 2 A more heart-breaking question. What is the probability that the grub tonight is good 1? Solution: an easy guess, nevertheless a rude reminder of reality. These are not the only terrible things that stare at you. Example 3 The chance of you encountering our beloved Mr. Leopard on your way back. Answer: This really looks like a no probability event, i.e. none of you or your immediate hand-shakers might have spotted one here, apart from being touched on the ear by a dark striped yellow rumor. However, can you really say that there is no chance. Example 4 The chance of you getting caught while ragging someone junior. Answer: I will rather be with the leopard. We have used the word chance couple of times, it is important to identify our existing views of this concept. Let us do the following exercise. Exercise 1 Take your notebook and write the word chance, and note down all possible meanings and a few sentences that you can conjure with it. (I know, you don t like this, but please make an attempt) Is chance an optimistic opportunity or one that plays against us? When we hear that Tendulkar chanced his arm against the seamers, who is sensing the opportunity? The further we dwell into these intricacies, we not only enter the philosopher s territory, but also wander away from the real answers. So we need a language which can quantify chances, that the games are played fair and square between the entities. Three, Two, Five, Start.. 1 There are worse questions, for example the chance of finding something else

2 1.1 Learning Cricket Cricket is a game of chance, we all know this (otherwise go watch Jadeja). Chances also bring out characters. Let us read the wall and ask the following questions, 1. Is Rahul Dravid a stable batsman? 2. Is Rahul Dravid a fairer batsman than Sachin Tendulkar? These questions are quite ambiguous, and an uninitiated may ask for photographs of these people. We can use statistics to understand and formalize these questions better. Statistics helps us to deduct relevant information from the book-keeping of past events, with an eye towards future, while allowing the existence of bookies and other gods of cricket, elements we can call rogue than random. Let us try to understand the first question above. Exercise 2 Table 1 shows some statistics from RD s illustrious career. Can you fill the missing entries in the table (Feel free to dig the web). Format Innings Runs Average Balls StrikeRate Test I class ?? Table 1: Rahul Dravid s Career Statistics (courtesy:tba) Statistics show that there is some sort of stability, that he has scored more than 5 runs several times before being dismissed. But as engineers, we need not worry about dead statistics, rather we engineer things for future. So let us imagine a player RDX (perhaps the robotic clone of RD), who will guard the wall from now on and never retires. Actually we can model our RDX to be stable in the following way (from our experience). The chance of RDX scoring a run is 1 5 for every ball, and this is true no matter what the asking-rate or team-demands are (imagine all that elegant drives laser-guided to the cover fielder). It is as if every ball is a new experiment. He is not fair in our sense, coz he is biased towards not-scoring. On the other hand, TDX (the avatar of SRT) is a fair player. The chance of him scoring runs in a given ball is 1 2 (I know this is a bit off the road from reality, but allow me to stick with it). If the run-demand is higher, he may score higher quanta of runs, but his chance of scoring any remains at.5 every future ball. Exercise 3 What is the chance that RDX explodes?, for him this means to score runs in the first 1 balls he faces. Our tale has more twists to it, actually we are also interested in questions like What is the chance that RDX has a hundred ball dry-run in future?. I guess this question is kind of new to you. You are more used to questions like what is the probability that RDX scores no run in the next ten balls? etc, which can be answered by simple counting. As opposed to this, the number of experiments is unlimited in the former question. So we need a more versatile theory than simple combinatorial probability dealing with a limited set of events. A comprehensive treatment of probability is an exhaustive course, usually administered under the ambit probability and measure in advanced graduate studies, and we are not 2

3 going to do that here. We are going to cover an engineering ground, or in cricketing terms, will tickle on in ones and twos, i.e. a lot of running to do. Let us have some more cricket. Let us play fair and devote some attention to the player TDX, before he became a full time M.P. Shown below is a graph of his performance over the years with the bat. The performance measure on Y axis is the runs scored between successive dismissals(on average), i.e. total runs divided by the number of dismissals that year. Yearly Batting Avg Figure 1: SRTs batting over the years (Yearly Averages Shown, Courtesy: TBA) Let us distill some useful information from the given data. One way of statistical inference is to construct histograms, i.e. make a chart with the fraction of years corresponding to a certain batting performance. However, in the current data, no batting average repeats itself, so each number will correspond to 1 year, or a fraction 1/24. This is not good book-keeping. Alternately, we can make blocks of 1 runs, and ask what fraction of the years did SRT s batting average happened to be in that block. For example, in Figure 2, SRT has a batting average between 4 and 5 for 25% of the years. You can try different block-sizes to make the chart, for example, intervals of 5 runs, and the resulting charts will fluctuate. This is also due to the insufficient data at our disposal. Nevertheless, the 1 run histogram that we constructed contains enough information to make some predictions about our maestro s future exploits with the bat. It is tempting to call the histogram as the distribution of batting performance. We will call it a bit more precisely as the mass distribution. In other words, it tells how a unit mass is distributed among different (non-overlapping) groups. We will reserve the word cumulative distribution (cdf in short) for a function which tells What fraction of total years did SRT perform below a given value?. Considering blocks of 1 runs as above, we can find the distribution from the mass-distribution, which is plotted in Figure 3. To help our understanding, we have also shown the mass distribution in the same figure. While plotting the distribution, we have used the convention that the leftmost point in an interval will correspond to the fraction of years below that interval. For example, the batting average was below 3 for 1 6 of the years and below 4 for a fraction 1 4. Clearly, the batting average was below 1 all these years. The terminology, right continuous is used 3

4 Fraction of Years Figure 2: Fraction of years in playing career with a given average Courtesy: TBA Fraction of Years Figure 3: Fraction of years in playing career below a given performance Courtesy: TBA 4

5 for such functions, i.e. in case of a step or jump, the function takes the value immediately after the jump. Exercise 4 Justify the name cumulative distribution function. Exercise 5 Can you identify the batsman who had the yearly batting distribution as shown in the figure below (consider only players with over 1 tests). 1 Fraction of Years Figure 4: Fraction of years in playing career below a given performance Courtesy: TBA Exercise 6 Plot the cdfs of yearly batting performance for 3 players of different nationality. Can you propose a measure of consistency to judge who is better. 5