Unit 9 - Statistics V3100018.001 UNIT 9 Continuous random...

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Statistics – V3100018.001 UNIT 9 – Continuous random variables Giuseppe Arbia , Catholic University of the Sacred Hearth, Roma, Italy 1
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2 In the discrete random variables all the characteristics are contained in the probability function f(x) for each value x of X Probability H T
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3 For continuous random variable (e.g. weight, height, waiting time) we cannot define a probability function for each value of the random variable because there are an infinite number of possible alternatives each with probability zero. Example 1: Prob (height = exactly 6 feet) = 1 / = 0 Example 2: Prob (waiting time = exactly 2.5 minutes) = 1 / = 0
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4 We can instead calculate the probability with which a continuous variable is observed in certian interval Examples: Prob(5,6< height< 5,8) Prob(2 < waiting time < 3) P(height > 5,8) P(waiting time < 1.30)
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5 In descriptive statistics we examined a continuous statistical variable through the distribution of the relative frequencies and graphical through the histogram. In such graphs, the sum of the relative frequencies (that is the sum of the area of the rectangles) is equal to 1.
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6 Height - Interval 5.0000 0.000 0.060 0.120 0.180 0.240 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 Height class Relative frequency Total surface area = 1 Surface area of a rectangle = frequency
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7 Height - Interval 2.5000 0.000 0.030 0.060 0.090 0.120 135.0 140.0 145.0 150.0 155.0 160.0 165.0 170.0 175.0 180.0 185.0 190.0 195.0 200.0 205.0 210.0 Height class Relative frequency
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8 Height - Interval 1.2500 0.000 0.015 0.030 0.045 0.060 135.00 138.75 142.50 146.25 150.00 153.75 157.50 161.25 165.00 168.75 172.50 176.25 180.00 183.75 187.50 191.25 195.00 198.75 202.50 206.25 210.00 Height class Relative frequency
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9 Height - Interval 0.6250 0.0000 0.0075 0.0150 0.0225 0.0300 135.000 138.750 142.500 146.250 150.000 153.750 157.500 161.250 165.000 168.750 172.500 176.250 180.000 183.750 187.500 191.250 195.000 198.750 202.500 206.250 210.000 Height class Relative frequency
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10 Height - Interval 0.3125 0.00000 0.00375 0.00750 0.01125 0.01500 135.0000 138.4375 141.8750 145.3125 148.7500 152.1875 155.6250 159.0625 162.5000 165.9375 169.3750 172.8125 176.2500 179.6875 183.1250 186.5625 190.0000 193.4375 196.8750 200.3125 203.7500 207.1875 Height class Relative frequency
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11 If we could use an infinite number of classes and an infinite number of observations, we could repeat this operation until the line connecting the upper side of each rectangle can be approximated by a continuous line. Height - Interval 5.0000 0.000 0.060 0.120 0.180 0.240 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 Height class Relative frequency 0.00000 0.00375 0.00750 0.01125 0.01500 Height class Height - Interval 0.3125 Such a curve is called probability density function
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