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# slide6 - Chapter 6: Continuous Random Variables and the...

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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

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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
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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
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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 . Mann E6 13/48

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6.2 The Normal Distribution Deﬁnition (Normal Distribution) A random variable X with probability density function f ( x ) = 1 2 πσ e - ( x - μ ) 2 2 σ 2 for - ∞ < x < has a normal distribution ( and is called a normal random variable ) with parameters μ and σ , where -∞ < μ < , and σ > 0. Also, E ( X ) = μ and V ( X ) = σ 2 Mann E6 14/48
6.2 The Normal Distribution Deﬁnition (Normal Distribution) A random variable X with probability density function f ( x ) = 1 2 πσ e - ( x - μ ) 2 2 σ 2 for - ∞ < x < has a normal distribution ( and is called a normal random variable

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## This note was uploaded on 02/05/2012 for the course ST 210 taught by Professor Wangs during the Fall '09 term at S. Alabama.

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slide6 - Chapter 6: Continuous Random Variables and the...

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