Unit 6a-1

Unit 6a-1 - Continuous Distributions Distributions...

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Continuous Continuous Distributions Distributions
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Continuous random variables Continuous random variables Are numerical variables whose values fall within a range or interval Are measurements Can be described by density curves
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Density curves Density curves Are always on or above on or above the horizontal axis Has an area exactly equal to one equal to one underneath it Often describes an overall distribution Describe what proportions proportions of the observations fall within each range of values
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Unusual density curves Unusual density curves Can be any shape Are generic continuous distributions Probabilities are calculated by finding the area under the finding the area under the curve curve
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1 2 3 4 5 .5 .25 P(X < 2) =   ( 29 25 . 2 25 . 2 = How do you find the area of a triangle?
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1 2 3 4 5 .5 .25 P(X = 2) = 0 P(X < 2) = .25 What is the area of a line segment?
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In continuous  distributions,         P( P( X X < < X X < < 2) 2)  are the  same answer. Hmmmm… Is this different than discrete distributions?
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1 2 3 4 5 .5 .25 P(X > 3) = P(1 < X < 3) = Shape is a trapezoid – How long are the bases? ( 29 2 2 1 h b b Area + = .5(.375+.5)(1)=.4375 .5(.125+.375)(2) =.5 b 2 = .375 b 1 = .5 h = 1
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1 2 3 4 0.25 0.50 P(X > 1) = .75 .5(2)(.25) = .25 (2)(.25) = .5
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1 2 3 4 0.25 0.50 P(0.5 < X < 1.5) = .28125 .5(.25+.375)(.5) = . 15625 (.5)(.25) = .125
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Special Continuous Distributions
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Uniform Distribution Uniform Distribution Is a continuous distribution that is evenly (or uniformly) distributed Has a density curve in the shape of a rectangle Probabilities are calculated by finding the area under the curve ( 29 12 2 2 2 a b b a x x - = + = σ μ endpoints of the uniform  distribution How do you find the area of a rectangle?
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4.98 5.04 4.92 The Citrus Sugar Company packs sugar in bags labeled 5 pounds. However, the packaging isn’t perfect and the actual weights are uniformly distributed with a mean of 4.98 pounds and a range of .12 pounds. a)Construct the uniform distribution above. How long is this   rectangle? What is the height of this rectangle? What shape does a uniform distribution have? 1/.12
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What is the probability that a randomly selected bag will weigh more than 4.97 pounds? 4.98 5.04 4.92 1/.12 P(X > 4.97) = .07(1/.12) = .5833 What is the length of the shaded region?
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Find the probability that a randomly selected bag weighs between 4.93 and 5.03 pounds. 4.98 5.04 4.92 1/.12 P(4.93<X<5.03) = .1(1/.12) = .8333 What is the length of the shaded region?
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The time it takes for students to drive to school is  evenly distributed with a minimum of 5 minutes and  evenly distributed with a minimum of 5 minutes and  a range of 35 minutes. a range of 35 minutes.
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This note was uploaded on 12/09/2011 for the course STATS 221 taught by Professor Nielson during the Fall '10 term at BYU.

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Unit 6a-1 - Continuous Distributions Distributions...

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