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lec12 - f(x)dx = 1 P(X = a = P(a X a = It follows that P(a...

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Continuous Random Variables All properties of discrete RVs have direct counterparts for coninuous RVs. One basic difference: summations used in the case of discrete RVs are replaced by integrals. Summing over (uncountable) infinite many values corresponds to an integral. For e.g.,we define a cumulative distribution function (cdf) as follows: Definition: CDF of a X is a continuous random variable: The function F X ( t ) := P ( X t ) is called the cumulative distribution function of X . The only difference to the discrete case is that the cdf of a continuous variable is not a stairstep function. 1 Properties of F X The following properties hold for the cumulative distribution function F X for random variable X . 0 F X ( t ) 1 for all t F X is monotone increasing, (i.e. if x 1 x 2 then F X ( x 1 ) F X ( x 2 ) .) lim t →−∞ F X ( t ) = 0 and lim t →∞ F X ( t ) = 1 . However, there is slight difference from the discrete case: Definition: Probability Density Function For a continuous variable X with
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