basicsofprobabilitytheory - 1 The basics of probability...

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1 The basics of probability theory E.R. Morey: BasicsOfProbabilityTheory.tex and .pdf September 8, 2010 1.1 What does probability mean? 1.1.1 What is the probability of some event? In the "street sense" of the word, probability of an event is a measure of likeliness: how likely is it that the event will happen - the more likely it is that the event will occur, the higher the probability num- ber. That is, if the likelihood of an event occurring increases, its probability increases. Note that probability is from probable . The guy on the street would likely put a few restrictions on this measure of likelihood. He would probably add: 1. The probability of an event cannot be less than zero 2. The probability of an event cannot be greater than 1 (greater than 100% ) 3. The probability of something happening is 1 4. And, the probability of nothing happening is zero. Us statisticans agree with the guy on the street that probability has these four properties. But, these are just conventions. For example, there is no reason "likelihood" could not be measured on a scale from ° 3 to 1 : 5 where ° 3 corresponds to certainty. On Mars they might have a di/erent range on probability. Note a few things about this street de°nition of probability. It is an exante concept, not an expost concept: once something has happened we know what happened (what event occurred). Once an event has occured, it is certain; it is not a random variable, and does not have a probability of occurance. 1
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Sampling and probability are two side of the same thing: Consider the question of whether it will rain in Boulder tomorrow as a function of what the weather in Boulder is today and the weather elsewhere. Let±s assume that the weather tomorrow is a function of the both of these observables, plus some random component. A sample is a realization of an experiment with uncertain outcomes, the realization of a stochastic data-generating process. That is, tomorrow±s weather is, today, a RV with some probability density function where that probability density function depends on the weather today both in Boulder and elsewhere. Tomorrow±s weather will be a draw from that distribution. Tomorrow, we might draw (sample) a sunny day, or we might draw (sample) a rainy day. Today, all we can ask is what is the probability that tomorrow will be sunny given today is bla, bla, bla. For example, consider a discrete distribution with three spikes: one for the probability of rain, one for sun, and one for snow, where the height of each spike depends on whether today was rain, sun or snow. (possiblly insert graph here, maybe create in excel, and use the excel macro that converts stu/ to .tex) What if the issue was tomorrow±s temperature. Imagine we assumed tem- perature tomorrow would be a random draw from a normal distribution whose mean was today±s temperature, or whose mean was a weighted average of today and yesterday±s temperatures.
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  • Spring '15
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