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By James H.C. Creighton

Welcome to new territory: A direction in likelihood versions and statistical inference. the concept that of chance isn't really new to you after all. you've gotten encountered it given that early life in video games of chance-card video games, for instance, or video games with cube or cash. and also you find out about the "90% probability of rain" from climate experiences. yet when you get past easy expressions of likelihood into extra sophisticated research, it really is new territory. and extremely overseas territory it's. you want to have encountered stories of statistical leads to voter sur­ veys, opinion polls, and different such experiences, yet how are conclusions from these reports received? how will you interview quite a few citizens the day earlier than an election and nonetheless make sure relatively heavily how HUN­ DREDS of millions of citizens will vote? that is records. you can find it very fascinating in this first path to work out how a effectively designed statistical examine can in achieving quite a bit wisdom from such greatly incomplete info. it truly is possible-statistics works! yet HOW does it paintings? by way of the top of this path you should have understood that and masses extra. Welcome to the enchanted forest.

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G) Does P(AIB) = P(BIA) imply that A and B are independent? (h) Show that P(AIB) = P(B IA) is equivalent to P(A) = P(B). 2 Before now we were using simple addition and multiplication rules for probabilities: "Or" means add. "And" means multiply. Of course, we warned you that these rules don't hold in complete generality. Let's explore these rules a bit. In what follows, let X be the number of dots on the top face of a fair die: (a) Under what conditions do these simple rules hold? (b) Which simple rule calculates P(2 :S X :S 5)?

6 In the text above we saw a loaded die that's less predictable than a fair die. (a) Is it true that any loaded die will be less predictable than a fair die? Explain. 2 - Parameters to Characterize a Probability Distribution 19 (b) Sketch a graph to illustrate part (a). 7 What about the unfair coin we looked at earlier which comes up heads 90% of the time? Compare it with a fair coin. (a) Is the number of heads on the unfair coin more dispersed about its mean than on a fair coin, or less so? As always, first try to guess.

You may be wondering how we can justify squaring the deviations. After all, that significantly changes their values. True, but the variance is used only in a comparative way to see that one distribution is more dispersed or less dispersed about its mean than another one. Because we'll always use the same procedure-squaring the deviations-it's valid for comparative purposes. So the variance by itself means nothing. 9167! We CAN ask what it means that our loaded die has a variance bigger than that of the fair die.

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