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RandomVariables - Outline Random Variables and Their...

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Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables Random Variables and Their Expected Values Michael Akritas Michael Akritas Random Variables and Their Expected Values
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Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables Random Variables and Their Distribution Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function The Expected Value of Discrete Random Variables The Expected Value in the Simplest Case General Definition for Discrete RVs Types of Random Variables Michael Akritas Random Variables and Their Expected Values
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Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Definition Let S be the sample space of some probabilistic experiment. A function X : S → R is called a random variable . Example 1. A unit is selected at random from a population of units. Thus S is the collection of units in the population. Suppose a characteristic (weight, volume, or opinion on a certain matter) is recorded. A numerical description of the outcome is a random variable. 2. S = { s = ( x 1 . . . , x n ) : x i R , i } , X ( s ) = i x i or X ( s ) = x , or X ( s ) = max { x 1 . . . , x n } . 3. S = { s : 0 s < ∞} (e.g. we may be recording the life time of an electrical component), X ( s ) = I ( s > 1500), or X ( s ) = s , or X ( s ) = log( s ). Michael Akritas Random Variables and Their Expected Values
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Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function A random variable X induces a probability measure on the range of its values, which is denoted by X ( S ). X ( S ) can be thought of as the sample space of a compound experiment which consists of the original experiment, and the subsequent transformation of the outcome into a numerical value. Because the value X ( s ) of the random variable X is determined from the outcome s , we may assign probabilities to the possible values of X . For example, if a die is rolled and we define X ( s ) = 1 for s = 1 , 2 , 3 , 4, and X ( s ) = 0 for s = 5 , 6, then P ( X = 1) = 4 / 6, P ( X = 0) = 2 / 6. The probability measure P X , induced on X ( S ) by the random variable X , is called the (probability) distribution of X . Michael Akritas Random Variables and Their Expected Values
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Outline Random Variables and Their Distribution The Expected Value of Discrete Random Variables Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function The distribution of a random variable is considered known if the probabilities P X (( a , b ]) = P ( a < X b ) are known for all a < b .
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