05_02 - Example (Uniform Distributions). If 1 , if < x ,...

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p. 5-11 Example (Uniform Distributions). If then f X ( x )= ½ 1 β α , if α <x β , 0 , otherwise, Some properties of expectation Expectation of Transformation. If Y = g ( X ), then provided that the integral converges absolutely. proof. ( homework ) E ( Y R −∞ y · f Y ( y ) dy = R −∞ g ( x ) · f X ( x ) dx, Expectation of Linear Function. E ( aX + b )= a · E ( X )+ b , since p. 5-12 Definition. If X has a pdf f X , then the expectation of X is also called the mean of X or f X and denoted by X , so that The variance of X is defined as and denoted by . The X is called the standard deviation . Some properties of mean and variance The mean and variance for continuous random variables have the same intuitive interpretation as in the discrete case. Var ( X ) = E ( X 2 ) – [ E ( X )] 2 Variance of Linear Function. Var ( aX + b )= a 2 · Var ( X ) Theorem. For a nonnegative continuous random variable X , Proof. μ X = E ( X R −∞ x · f X ( x ) dx. Var ( X E [( X μ x ) 2 ]= R −∞ ( x μ X ) 2 · f X ( x ) dx, σ 2 X E ( X R 0 1 F X ( x ) dx = R 0 P ( X>x ) dx. E ( X R 0 x · f X ( x ) dx = R 0 ¡R x 0 1 dt ¢ f X ( x ) dx = R 0 R x 0 f X ( x ) dt dx = R 0 R t f X ( x ) dx dt = R 0 1 F X ( t ) dt. 0
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p. 5-13 Reading : textbook, Sec 5.1, 5.2, 5.3, 5.7 Example (Uniform Distributions) • Uniform Distribution Some Common Continuous Distributions Summary for X ~ Uniform(  ) Pdf: Cdf: Parameters: < < < Mean: E ( X )=( + )/2 Variance: Var ( X )= (  ) 2 /12 F ( x )= 0 , if x α , ( x α ) / ( β α ) , α <x β , 1 , x> β .
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05_02 - Example (Uniform Distributions). If 1 , if &lt; x ,...

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