Chap8-Continuous-Probability-Distributions

Chap8-Continuous-Probability-Distributions - 1 CHAPTER 8:...

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Unformatted text preview: 1 CHAPTER 8: CONTINUOUS PROBABILITY DISTRIBUTIONS Stats & Prob. for Bus. Mgmt (Stat1100) Jochem Chap 8: Continuous Prob. Distrib. b Overview b 1. Probability Density Functions b 2. Continuous Probability Distributions s 2.1 The Uniform Distribution 2 s 2.2 The Normal Distribution s 2.3 The Exponential Distribution s 2.4 The Student t Distribution s 2.5 The Chi-Squared Distribution s 2.6 The F Distribution b 3. Excel Commands b 1. Probability Density Functions b Recall that in Chap 7 we discussed discrete versus continuous Random Variables (RVs). We said that discrete RVs have countable many values, while ntinuous ones have uncountable many. Chap 8: Continuous Prob. Distrib. 3 continuous ones have uncountable many. X = Sum of 2 dices = Y=time to finish = writing an exam 2: 1 3: 2 4: 3 5: 4 6: 5 7: 6 8: 5 9: 4 10:3 11:2 12:1 30 min 30 min 1 sec? or 30 min 1/10 sec? or 30 min 1/100 sec? 31 min . To count the outcomes we need to know what is the next outcome after, say, 30 min. Discrete RV Continuous RV b 1. Probability Density Functions b Implications of an RV being continuous s 1. We cannot create a probability or frequency table (the table would never end). Chap 8: Continuous Prob. Distrib. 4 s 2. Recall that if we add up all the probabilities of a probability distribution, they need to equal 1. If we have however uncountable many outcomes, each single outcome must have a probability of virtually 0! B This is referred to as Point probabilities are zero. b 1. Probability Density Functions b Step 1: As a result of (1) we can only draw histograms with intervals for continuous RVs. Chap 8: Continuous Prob. Distrib. 5 b 1. Probability Density Functions b Step 1: As a result of (1) we can only draw histograms with intervals for continuous RVs. b Step 2: Lets rescale the histogram by (N*interval size), Chap 8: Continuous Prob. Distrib. 6 so that we have that the sum of bar areas equals to 1. =40/(200*5) b 1. Probability Density Functions b Step 3: Finally, lets reduce the size of the intervals from 15 minutes further and further. If we connect the tops of all very very slim bars, well get a smooth line. Chap 8: Continuous Prob. Distrib. 7 Enclosed area = 1 This is called the probability density of a continuous random variable. If f is a function that approximates that curve, then f(x) is called the probability density function. b 1. Probability Density Functions b Step 3: Finally, lets reduce the size of the intervals from 15 minutes further and further. If we connect the tops of all very very slim bars, well get a smooth line. Chap 8: Continuous Prob. Distrib. 8 This area is the probability that the exam time is between 15 min & 30 min....
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This note was uploaded on 03/26/2012 for the course STAT 1100 taught by Professor Chiappetta during the Spring '08 term at Pittsburgh.

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Chap8-Continuous-Probability-Distributions - 1 CHAPTER 8:...

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