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10_3_11

10_3_11 - Mean and Variance of Continuous Random Variables...

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Mean and Variance of Continuous Random Variables; Continuous Uniform Distribution Andrew Liu October 3, 2011 Andrew Liu Textbook section: 4-4, 4-5

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Mean and Variance of Continuous Random Variables Comparison between discrete and continuous random variables Discrete random variable Continuous random variable Mean μ = X x xf ( x ) μ = Z -∞ xf ( x ) dx Var σ 2 = E ( X - μ ) 2 = X x ( x - μ ) 2 f ( x ) = E ( X ) 2 - μ 2 = X x x 2 f ( x ) - μ 2 σ 2 = E ( X - μ ) 2 = Z -∞ ( x - μ ) 2 f ( x ) dx = E ( X ) 2 - μ 2 = Z -∞ x 2 f ( x ) dx - μ 2 Andrew Liu Textbook section: 4-4, 4-5
Mean and Variance of Continuous Random Variables Comparison between discrete and continuous random variables Discrete random variable Continuous random variable Mean μ = X x xf ( x ) μ = Z -∞ xf ( x ) dx Var σ 2 = E ( X - μ ) 2 = X x ( x - μ ) 2 f ( x ) = E ( X ) 2 - μ 2 = X x x 2 f ( x ) - μ 2 σ 2 = E ( X - μ ) 2 = Z -∞ ( x - μ ) 2 f ( x ) dx = E ( X ) 2 - μ 2 = Z -∞ x 2 f ( x ) dx - μ 2 Rule of thumb: For the mean and variance of continuous random variables 1 Replace probability mass function of discrete random variables by density function. 2 Replace summation by integral. Andrew Liu Textbook section: 4-4, 4-5

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Mean and Variance of Continuous Random Variables Comparison between discrete and continuous random variables Discrete random variable Continuous random variable Mean μ = X x xf ( x ) μ = Z -∞ xf ( x ) dx Var σ 2 = E ( X - μ ) 2 = X x ( x - μ ) 2 f ( x ) = E ( X ) 2 - μ 2 = X x x 2 f ( x ) - μ 2 σ 2 = E ( X - μ ) 2 = Z -∞ ( x - μ ) 2 f ( x ) dx = E ( X ) 2 - μ 2 = Z -∞ x 2 f ( x ) dx - μ 2 Rule of thumb: For the mean and variance of continuous random variables 1 Replace probability mass function of discrete random variables by density function. 2 Replace summation by integral. Intuitive understanding of mean – Consider a stick, the mean is the balance point of the stick. Andrew Liu Textbook section: 4-4, 4-5
Mean and Variance – Examples E.g.1 Suppose f ( x ) = 0 . 125 x for 0 < x < 4. Determine the mean and variance of X . Andrew Liu Textbook section: 4-4, 4-5

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E.g.1 Suppose f ( x ) = 0 . 125 x for 0 < x < 4. Determine the mean and variance of X . Solution:
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10_3_11 - Mean and Variance of Continuous Random Variables...

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