Continuous Random Variables (1)

Continuous Random Variables (1) - Continuous Random...

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Sections 3.4 and 3.5 Continuous Random Variables Sections 3.4 to 3.5 Continuous Random Variable A continuous random variable is one for which the outcome can be any value in an interval of the real number line. Examples Let Y = length in mm Let Y = time in seconds Let Y = temperature in ºC head2right We don’t calculate P(Y = y) , we calculate P(a < Y < b) , where a and b are real numbers. head2right For a continuous random variable o P(Y = y) = 0 . head2right Continuous Random Variables head2right The probability density function (pdf) when plotted against the possible values of Y forms a curve. The area under an interval of the curve is equal to the probability that Y is in that interval. o The entire area under a probability density curve for a continuous random variable is always equal to 1. 5 10 15 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 x y
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Sections 3.4 and 3.5 Properties Properties of a Probability Density Function (pdf) 1. g1858g4666g1877g4667g34100 for all possible intervals of y—recall that probabilities cannot be negative. 2. g1516 g1858g4666g1877g4667g1856g1877 g2879 g34041 —the maximum range of any continuous variable is g4666g3398 ,g3397 g4667 and the probability over the entire range (sample space) must be 1. 3. If g1877 g2868 is a specific value of interest, then the cdf is defined as g1832g4666g1877 g2868 g4667g3404 g1842g4666g1851 g3409 g1877 g2868 g4667g3404 g3505 g1858g4666g1877g4667g1856g1877 g3052 g3116 g2879 4. If g1877 g2869 and g1877 g2870 are specific values of interest then g1842g4666g1877 g2869 g3409g1851 g3409g1877 g2870 g4667 g3404g3505 g1858g4666g1877g4667g1856g1877 g3052 g3118 g3052 g3117 g3404g1832g4666g1877 g2870 g4667g3398g1832g4666g1877 g2869 g4667 5. Other probability rules also apply! 6. If g1877 g2868 is specific values of interest then g1842g4666g1851 g3404 g1877 g2868 g4667g3404 0
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Sections 3.4 and 3.5 Expected Value and Variance Recall that the expected value is a weighted average over all possible values of g1877 . g2020 g3404 g1831g4666g1851g4667 g3404g3505 g1877g1858g4666g1877g4667g1856g1877 g2879 And the variance, g2026 g2870 g3404g1848g1853g1870g4666g1851g4667 g3404g3505 g1877 g2870 g1858g4666g1877g4667g1856g1877 g2879 g3398g2020 g2870
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Sections 3.4 and 3.5 0 5 10 15 20 25 30 0.020 0.025 0.030 0.035 0.040 0.045 wait time f(y) The Uniform Distribution g1858g4666g1877g4667g3404 1 g1854g3398g1853 g1858g1867g1870 g1853 g3409 g1877 g3409 g1854 g2020 g3404g1831g4666g1851g4667 g3404 g1853g3397g1854 2 g2026 2 g3404 g4666g1854g3398g1853g4667 2 12 Example. A bus arrives at a bus stop every 30 minutes. If a person arrives at the bus stop at a random time, what is the probability that the person will have to wait less than 10 minutes for the next bus? Let Y = wait time in minutes. Since the arrival time is random, someone is as likely to arrive 1 minute before a bus arrives as 2 minutes, as 3 minutes, etc. head2right What is the probability a person will wait less than 10 minutes? head2right What is the probability that a person will have to wait between 5 and 25 minutes?
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Sections 3.4 and 3.5 head2right What is the probability that a person will have to wait more than 25 minutes? head2right What is the probability that a person will have to wait 25 minutes or more?
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