introduction-probability.pdf

Example 3 now the situation is a bit more complicated

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Example 3. Now the situation is a bit more complicated: Somebody has two dice and transmits to you only the sum of the two dice. The set of all outcomes of the experiment is the set Ω := { (1 , 1) , ..., (1 , 6) , ..., (6 , 1) , .... , (6 , 6) } 5
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6 CONTENTS but what you see is only the result of the map f : Ω R defined as f (( a, b )) := a + b, that means you see f ( ω ) but not ω = ( a, b ). This is an easy example of a random variable f , which we introduce later. Example 4. In Example 3 we know the set of states ω Ω explicitly. But this is not always possible nor necessary: Say, that we would like to measure the temperature outside our home. We can do this by an electronic ther- mometer which consists of a sensor outside and a display, including some electronics, inside. The number we get from the system might not be correct because of several reasons: the electronics is influenced by the inside temper- ature and the voltage of the power-supply. Changes of these parameters have to be compensated by the electronics, but this can, probably, not be done in a perfect way. Hence, we might not have a systematic error, but some kind of a random error which appears as a positive or negative deviation from the exact value. It is impossible to describe all these sources of randomness or uncertainty explicitly. We denote the exact temperature by T and the displayed temperature by S , so that the difference T - S is influenced by the above sources of uncertainty. If we would measure simultaneously, by using thermometers of the same type, we would get values S 1 , S 2 , ... with corresponding differences D 1 := T - S 1 , D 2 := T - S 2 , D 3 := T - S 3 , ... Intuitively, we get random numbers D 1 , D 2 , ... having a certain distribution. How to develop an exact mathematical theory out of this? Firstly, we take an abstract set Ω. Each element ω Ω will stand for a specific configuration of our sources influencing the measured value. Secondly, we take a function f : Ω R which gives for all ω Ω the difference f ( ω ) = T - S ( ω ). From properties of this function we would like to get useful information of our thermometer and, in particular, about the correctness of the displayed values. To put Examples 3 and 4 on a solid ground we go ahead with the following questions: Step 1: How to model the randomness of ω , or how likely an ω is? We do this by introducing the probability spaces in Chapter 1. Step 2: What mathematical properties of f we need to transport the ran- domness from ω to f ( ω )? This yields to the introduction of the random variables in Chapter 2.
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CONTENTS 7 Step 3: What are properties of f which might be important to know in practice? For example the mean-value and the variance, denoted by E f and E ( f - E f ) 2 . If the first expression is zero, then in Example 3 the calibration of the ther- mometer is right, if the second one is small the displayed values are very likely close to the real temperature. To define these quantities one needs the integration theory developed in Chapter 3.
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