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Chapter 4 - Continuous Random Variables Possible values are...

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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
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Expected value ( μ ) For a continuous variable x, the expected value is μ = ( ) * ( ) X E x x f x dx = Variance ( σ 2 29 σ 2 = 2 2 ( ) [ ] ( [ ]) Var x E x E x = - where 2 2 [ ] * ( ) X E x x f x dx = Examples: (1) Let f(x) = 1.5x 2 for –1 < x < 1. a. Prove that this is a valid probability density function.
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