Continuous Random Variables (1)

# Continuous Random - Continuous Random Variables Sections 3.4 to 3.5 Continuous Random Variable A continuous random variable is one for which the

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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 h We don’t calculate P(Y = y) , we calculate P(a < Y < b) , where a and b are real numbers. h For a continuous random variable o P(Y = y) = 0 . h Continuous Random Variables h 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. gG±² ³ 0 for all possible intervals of y—recall that probabilities cannot be negative. 2. ´ gG±²µ± · 1 —the maximum range of any continuous variable is ² and the probability over the entire range (sample space) must be 1. 3. If ± º is a specific value of interest, then the cdf is defined as »G± º ² · ¼G½ ¾ ± º ² · ¿ gG±²µ± À Á 4. If ± Â and ± Ã are specific values of interest then ¼G± Â ¾ ½ ¾ ± Ã ² · ¿ gG±²µ± À Ä À Å · »G± Ã ² ¸ »G± Â ² 5. Other probability rules also apply! 6. If ± º is specific values of interest then ¼G½ · ± º ² · 0
Sections 3.4 and 3.5 Expected Value and Variance Recall that the expected value is a weighted average over all possible values of g . G ± ²³´µ ± ¶ g·³gµ¸g ¹ And the variance, º » ± ¼½¾³´µ ± ¶ g » ·³gµ¸g ¹ ¿ G »

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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 gG±² ³ 1 ´ µ ¶ g·¸ ¶ ¹ ± ¹ ´ º ³ »G¼² ³ ¶ ½ ´ 2 ¾ 2 ³ G´ µ ¶² 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. h What is the probability a person will wait less than 10 minutes? h What is the probability that a person will have to wait between 5 and 25 minutes?
Sections 3.4 and 3.5 h What is the probability that a person will have to wait more than 25 minutes? h What is the probability that a person will have to wait 25 minutes or more? h

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## This note was uploaded on 10/03/2011 for the course STAT E509 taught by Professor Wheatley during the Spring '10 term at South Carolina.

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Continuous Random - Continuous Random Variables Sections 3.4 to 3.5 Continuous Random Variable A continuous random variable is one for which the

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