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10_5_11

# 10_5_11 - Continuous Uniform Distribution Exponential...

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Unformatted text preview: Continuous Uniform Distribution; Exponential Distribution Andrew Liu October 5, 2011 Andrew Liu Textbook section: 4-5, 4-8 Review; Mean and Variance of Continuous Random Variables Mean and variance of functions of random variables Suppose that Y = h ( X ). (For example, Y = 3 X 2- 4.) Let f ( x ) denote the density function of X . Andrew Liu Textbook section: 4-5, 4-8 Review; Mean and Variance of Continuous Random Variables Mean and variance of functions of random variables Suppose that Y = h ( X ). (For example, Y = 3 X 2- 4.) Let f ( x ) denote the density function of X . Discrete random variable Continuous random variable Mean E ( Y ) = X i h ( x i ) P ( X = x i ) E ( Y ) = Z ∞-∞ h ( x ) f ( x ) dx Var V ( Y ) = X i h ( x i ) 2 P ( X = x i )- [ E ( Y )] 2 V ( Y ) = Z ∞-∞ h 2 ( x ) f ( x ) dx- [ E ( Y )] 2 Andrew Liu Textbook section: 4-5, 4-8 Mean and Variance – Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 ≤ x ≤ 20. Determine the mean and variance of Y = X 2 . Andrew Liu Textbook section: 4-5, 4-8 Mean and Variance – Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 ≤ x ≤ 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Andrew Liu Textbook section: 4-5, 4-8 Mean and Variance – Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 ≤ x ≤ 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Z ∞-∞ h ( x ) f ( x ) dx = Andrew Liu Textbook section: 4-5, 4-8 Mean and Variance – Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 ≤ x ≤ 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Z ∞-∞ h ( x ) f ( x ) dx = Z 20 ( x 2 * . 05) dx = 0 . 05 * x 3 3 20 . Andrew Liu Textbook section: 4-5, 4-8 Mean and Variance – Examples E.g.1 Suppose f ( x ) = 0 . 05 for 0 ≤ x ≤ 20. Determine the mean and variance of Y = X 2 . Solution: E ( Y ) = Z ∞-∞ h ( x ) f ( x ) dx = Z 20 ( x 2 * . 05) dx = 0 . 05 * x 3 3 20 . V ( Y ) = Z ∞-∞ h ( x ) 2 f ( x )- [ E ( Y )] 2 = Z 20 ( x 4 * . 05) dx- [ E ( Y )] 2 = . 05 * x 5 5 20- [ E ( Y )] 2 . Andrew Liu Textbook section: 4-5, 4-8 Continuous Uniform Distribution The Simplest Continuous Distribution Uniform Distribution: Density Function Uniform Distribution: Cumulative Distribution Function Andrew Liu Textbook section: 4-5, 4-8 Continuous Uniform Distribution – Density Function, Mean and Variance Density function A continuous random variable X is called a continuous uniform random variable if it has the the following probability density function. f ( x ) = 1 ( b- a ) , a ≤ x ≤ b . Andrew Liu Textbook section: 4-5, 4-8 Continuous Uniform Distribution – Density Function, Mean and Variance Density function A continuous random variable X is called a continuous uniform random variable if it has the the following probability density function....
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10_5_11 - Continuous Uniform Distribution Exponential...

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