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Unformatted text preview: Continuous Random Variables Possible values are any value in some interval of the real number line. The number of possible values is uncountably infinite Density function f(x) is the probability density function (PDF) probability = area under a curve ( ) ( ) b a P a x b f x dx = has two properties: 1) f(x) > 0 (non-decreasing function) 2) (integrating over region for which x is defined results in 1) If X is continuous, then P(x 1 < X < x 2 ) = P(x 1 < X < x 2 ). The result is the same whether we use greater than or greater than/equal to. WHY? Cumulative Distribution Function For a continuous random variable, X, the CDF ( ) ( ) ( ) x F X P X x f t dt- = = Note: f(x) is the derivative of F(x) w.r.t. x. - = 1 ) ( dx x f Expected value ( ) For a continuous variable x, the expected value is = ( ) * ( ) X E x x f x d x = Variance ( 2 29 2 = 2 2 ( ) [ ] ( [ ]) Var x E x E x =- where 2 2 [ ] * ( ) X E x x f x d x = Examples: (1) Let f(x) = 1.5x 2 for 1 < x < 1....
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