# lec8 - Statistics of R.V.s Expectation The expected value...

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Statistics of R.V.s Expectation The expected value of a function h ( X ) is defined as E [ h ( X )] := i h ( x i ) · p X ( x i ) . The most important version of this is the case h ( x ) = x : E [ X ] = i x i · p X ( x i ) =: μ E [ X ] is usually denoted by the symbol μ . The expected value of a random variable E [ X ] is a measure of the average value of the possible values of the random variable. We see that it is actually a weighted average of the values of X , weighted by the point masses p X 1

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Example Toss a Die Toss a fair die, and denote by X the number of spots on the upturned face. What is the expected value for X ? The probability mass function of X is p X ( i ) = 1 6 for all i ∈ { 1 , 2 , 3 , 4 , 5 , 6 } . Therefore, using the definition E [ X ] = 6 summationdisplay i =1 ip X ( i ) = 1 · 1 6 + 2 · 1 6 + 3 · 1 6 + 4 · 1 6 + 5 · 1 6 + 6 · 1 6 = 3 . 5 . This formula shows that E ( X ) is also the center of gravity of masses p X placed at corresponding points x . 2
Statistics of R.V.s Variance A second common measure for describing a random variable is a measure of how far its values are spread out. The variance of a random variable X is defined as: V ar [ X ] := E [( X E [ X ]) 2 ] = i ( x i E [ X ]) 2 · p X ( x i ) V ar [ X ] is usually denoted by the symbol σ 2 The variance is measured in squared units of X .

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