handout6 - Notes Chapter 6 Continuous Random Variables and...

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Chapter 6: Continuous Random Variables and the Normal Distributions Bin Wang [email protected] Department of Mathematics and Statistics University of South Alabama Mann E6 1/48 6.1 Continuous Probability Distribution A continuous random variable can assume any value over an interval or intervals. p.m.f won’t work for continous random varibales. Mann E6 2/48 Mann E6 3/48 Mann E6 4/48 Notes Notes Notes Notes
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Mann E6 5/48 Probability density function The probability distribution of a continuous random variable possesses the following two characteristics: 1 The probability that x assumes a value in any interval lies in the range 0 to 1; 2 The total probability of all the (mutually exclusive) intervals within which x can assume a value is 1.0. Mann E6 6/48 Mann E6 7/48 Mann E6 8/48 Notes Notes Notes Notes
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Mean and Variance Definition (Mean and Variance) Suppose X is a continuous random variable with probability density function f ( x ). The mean or expected value of X , denoted as μ or E ( X ), is μ = E ( X ) = Z -∞ xf ( x ) dx The variance of X , denoted as V ( X ) or σ 2 , is σ 2 = V ( X ) = Z -∞ ( x - μ ) 2 f ( x ) dx = Z -∞ x 2 f ( x ) dx - μ 2 The standard deviation of X is σ = [ V ( X )] 1 / 2 .
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