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Chap8-Continuous-Probability-Distributions

Chap8-Continuous-Probability-Distributions - 1 Chap 8...

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1 CHAPTER 8: CONTINUOUS PROBABILITY DISTRIBUTIONS Stats & Prob. for Bus. Mgmt (Stat1100) Jochem Chap 8: Continuous Prob. Distrib. box2 Overview box5 1. Probability Density Functions box5 2. Continuous Probability Distributions square6 2.1 The Uniform Distribution 2 square6 2.2 The Normal Distribution square6 2.3 The Exponential Distribution square6 2.4 The Student t Distribution square6 2.5 The Chi-Squared Distribution square6 2.6 The F Distribution box5 3. Excel Commands box2 1. Probability Density Functions box5 Recall that in Chap 7 we discussed discrete versus continuous Random Variables (RVs). We said that discrete RVs have countable many values, while continuous ones have uncountable many. Chap 8: Continuous Prob. Distrib. 3 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 box2 1. Probability Density Functions box5 Implications of an RV being continuous square6 1. We cannot create a probability or frequency table (the table would never end). Chap 8: Continuous Prob. Distrib. 4 square6 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! barb2right This is referred to as “ Point probabilities are zero.
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box2 1. Probability Density Functions box5 Step 1: As a result of (1) we can only draw histograms with intervals for continuous RVs. Chap 8: Continuous Prob. Distrib. 5 box2 1. Probability Density Functions box5 Step 1: As a result of (1) we can only draw histograms with intervals for continuous RVs. box5 Step 2: Let’s 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) box2 1. Probability Density Functions box5 Step 3: Finally, let’s reduce the size of the intervals from 15 minutes further and further. If we connect the tops of all very very slim bars, we’ll 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. box2 1. Probability Density Functions box5 Step 3: Finally, let’s reduce the size of the intervals from 15 minutes further and further. If we connect the tops of all very very slim bars, we’ll 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. 15min 30min This is called the probability density of a continuous random variable.
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