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MATH 226 - 10.17.07

MATH 226 - 10.17.07 - If X is continuous F(x = -∞ x...

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MATH 226 – October 17, 2007 Continuous Random Variables Probability density Function (PDF) o A function of all reals with the properties f(x) ≥ 0 -∞ f(x)dx = 1 o for a continuous random variable x with pdf f(x) we say: P(a ≤ x ≤ b) = the integral from a to b of ∫ a b f(x)dx Example 26 o Cotter pin length = 6 + y cm o f(y) = k(y + y 2 ), 0 ≤ y ≤ 2 o Find P(y ≥ 1) = P(1 ≤ y ≤ ∞) o = ∫ 1 f(y)dy = ∫ 1 2 f(y)dy = ∫ 1 2 k(y+y 2 )dy o To determine “k” 0 2 k(y + y 2 )dy k = 3/14 o = k(23/6) = (3/14)*(11/6) = 11/28 The cumulative distribution function (CDF) is o F(x) = P(X ≤ x) o
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Unformatted text preview: If X is continuous: F(x) = ∫-∞ x f(t)dt Find the cdf for the variable in problem 26 o F(y) = P(Y ≤ y) = { 0 for y ≤ 0 } { ∫ y (3/14)(t + t 2 )dt for 0 < y < 2 } { 1 for y ≥ 2 } o F(y) = P(Y ≤ y) = { 0 for y ≤ 0 } { (3y 2 + 2y 3 )/28 for 0 < y < 2 } { 1 for y ≥ 2 } The expected value of a continuous random variable is o EX = ∫-∞ ∞ xf(x)dx The variance is o Var(x) = ∫-∞ ∞ (x – μ) 2 f(x)dx = EX 2 – (EX) 2 Example 24 o A. Mean = ½ Var(x) = 1/(2*sqrt(3)) = .289 o B. o C. 1 - .995 = .005...
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