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Continuous Probability Distribution (Lesson - 03/C) Dealing with Uncertainty on a Grand Scale Continuous Distributions • In discrete random variable distributions, there is always a gap between the points of a distribution. For example: the observable numbers on a dice roll can only be 0, 1, 2, 3, 4, 5, or 6 but NOT 4.2 or 3.5, etc. • In a continuous distribution, there are no gaps between values Dr. C. Ertuna 2 Continuous Distributions • Discrete 0.3 Continuous 0.3 0.25 0.25 0.2 0.2 0.15 p(x) 0.15 f(x) 0.1 0.1 0.05 0.05 0 0 0 1 2 3 4 5 6 7 8 9 10 -0.05 X 0 1 2 3 4 5 6 7 8 9 10 X The distribution of a CRV is characterised by a probability density function f(x) Dr. C. Ertuna 3 Continuous Distributions • • • • • • • Uniform Distribution Normal Distribution z-Distribution t-Distribution F-Distribution Chi-Square Distribution etc. Dr. C. Ertuna 4 Continuous Distributions • The probability density function f(x) must satisfy two conditions; 1 f(x) 0 (i.e. non negative) 2 The total area under the curve is 1 • As the area of a line is zero, the specific value of any x value is always 0 Dr. C. Ertuna 5 Continuous Distributions (Cont.) There are 3 basic types of parameters for Continuous Distributions: – Shape (some may have more than one shape parameter) – Scale (specifies the size along the horizontal axis - contract/expend) – Location (specifies the location relative to zero on the horizontal axis). Dr. C. Ertuna 6 Uniform Distribution The sales of a company varies between $1,000 and 2,000 with equal probability. What is the the probability that the sales revenue will be less than or equal to $1,300? To answer this question we need to know how to put in use the Uniform Distribution. Dr. C. Ertuna 7 The Uniform Distribution • Could be used to model: – when little knowledge about a random variable is available. • Characteristics: – between minimum (a) and maximum (b) – outcomes are equally likely Dr. C. Ertuna 8 Uniform Distribution • A uniform distribution occurs when the values of a random variable are distributed evenly across the domain of the variable 0.12 0.1 0.08 0.06 0.04 0.02 0 0 1 2 3 4 5 Dr. C. Ertuna 6 7 8 9 9 Uniform Distribution (Cont.) a= b= f(x) = F(x) = F(x) = Minimum x = Specified Value Maximum = 1 / (b-a) if a ≤ x ≥ b Cumulative probability observing at most ( x ) = (x-a) / (b-a) if a ≤ x ≥ b =1 if x ≥ b =0 if x ≤ a Expected value = (a+b)/2 ; Var = [(b-a)^2] / 12 Dr. C. Ertuna 10 Example: Uniform Distribution The sales of a company varies uniformly between $1,000 and $2,000. What is the probability that the sales revenue will be less than or equal to $1,300? Specified Value Minimum Value Maximum Value x= a= b= The probability that the sales revenue will be $1,300 or less is F(x) 30% Dr. C. Ertuna 1,300 1,000 2,000 11 Excel: Uniform Distribution 1/A B C D E F G H 2 Specified Value x = 1,300 Specified Revenue 3 Minimum Value a = 1,000 Minimum Sales Revenue 4 Maximum Value b = 2,000 Maximum Sales Revenue 5 6 Prob getting ≤ x 7 8 =(D2-D3)/(D4-D3) F(x) = 30.00% probability that sales will be x or less Dr. C. Ertuna 12 The Normal Distribution The customer’s demand is normally distributed with a mean of 750 units/month and a standard deviation of 100 units/month What is the probability that the demand will exceed 900 units/month? To answer this question we need to know how to put in use the Normal Distribution. Dr. C. Ertuna 13 The Normal Distribution • The most important distribution that you will learn about this semester is the normal distribution • There are several characteristics that make the normal distribution very important for statisticians It is an excellent approximation to many populations. (Adult human heights and weights). Dr. C. Ertuna 14 Normal Distribution • Could be used to model: – many natural phenomena, – deviation from specifications (quality control). • Characteristics: – Symmetrical and bell-shaped – Most observations are close to the mean with gradually fewer further away – Can be determined entirely by the values of and Dr. C. Ertuna 15 Normal Distribution (Cont) • A typical normal distribution; It could completely be described by μ and σ Dr. C. Ertuna 16 Normal Distribution (Cont) • Two normal distributions with same σ but different μ Dr. C. Ertuna 17 Normal Distribution (Cont) • Two normal distributions with same μ but different σ Dr. C. Ertuna 18 Normal Distribution (Cont.) x= Specified Value μ= Mean σ2 = Variance f(x) = e^[-(x-μ)2/2σ2] / (2π σ2)1/2 F(x) = Cannot be expressed mathematically Dr. C. Ertuna 19 Normal Distribution (Cont) • For a normal distribution, the following rules hold true; • 68.3 % of all observations lie within 1 standard deviation of the mean • 95.4 % of all observations lie within 2 standard deviations of the mean • 99.7 % of all observations lie within 3 standard deviations of the mean Dr. C. Ertuna 20 Normal Distribution (Cont.) 68.3 % of all observations lie within 1 standard deviation of the mean Dr. C. Ertuna 21 Normal Distribution (Cont.) NORMDIST(x, mean, std_dev, cumulative) x= = = Specified value Standard deviation mean x Std_dev mean Expected value = E(x) = ; Var = 2 Dr. C. Ertuna 22 Example: Normal Distribution The customer’s demand is normally distributed with a mean of 750 units/month and a standard deviation of 100 units/month. What is the probability that the demand will exceed 900? Specified Value x= Mean μ= Standard Deviation (=square root of variance) σ = The probability that the demand will be more than 900 units is σ Dr. C. Ertuna 900 750 100 F(x) 6.68% 23 Excel: Normal Distribution 1/A B C D E F G H 2 Specified Value x = 900 Specified Demand 3 Mean μ = 750 Mean Demand 4 Standard_dev σ = 100 Standard Deviation of Demand 5 6 7 Prob getting > x 8 =NORMDIST(D2,D3,D4,TRUE) = 0.933 F(x) = 6.68% probability that demand will be more than x Dr. C. Ertuna 24 The Standard Normal Distribution • A special case of the normal distribution, the standard normal distribution has a mean of 0 and a standard deviation of 1 0.5 0.4 0.3 0.2 4 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 0 -3 0.1 -4 f(z) • The corresponding standard random variable is denoted by Z 0.6 Z Dr. C. Ertuna 25 Standard Normal Distribution (Cont) • Any normal distribution can be converted to a The Standard Normal Distribution, simply by converting it’s mean to 0 and it’s standard deviation to 1. ie Subtracting from each observation and dividing by . Z X Dr. C. Ertuna 26 How to Read a Box-and-Whisker Plot Dr. C. Ertuna 27 Exponential Distribution The mean time to fail of a light bulb is 12,000 hours. What is the probability that the light bulb will fail before 5,000 hours? To answer this question we need to know how to put in use the Exponential Distribution. Dr. C. Ertuna 29 Exponential Distribution • Could be used to model: – events that recur randomly over time, – customer arrivals (to a service system). • Characteristics: – Bounded below zero – Greatest density at 0 – Only scale parameter λ, no shape or location parameters Dr. C. Ertuna 30 Exponential Distribution (Cont.) EXPONDIST(x, lambda, cumulative) Specified value = 1/ mean f(x) = ex -x F(x) = 1- e x= x lambda x >= 0 x >= 0 E(x) = 1/ ; Var = (1/)2 Dr. C. Ertuna 31 Example: Exponential Distribution The mean time to fail of a light bulb is 12,000 hours. What is the probability that the light bulb will fail before 5,000 hours? Specified Value Mean Lambda = 1 / mean x= μ= = The probability that the light bulb will fail before 5,000 hours is F(x) 34.08% Dr. C. Ertuna 5000 12000 1/12000 32 Excel: Exponential Distribution 1/A B C D 2 Specified Value x = 5,000 3 Lambda = E F G H Specified Demand 0.00008 Mean Demand 4 5 6 Prob getting < x 7 8 =EXPONDIST(D2,D3,TRUE) F(x) = 34.08% probability that the bulb will fail before than x Dr. C. Ertuna 33 Summary Probability distributions help managers in: • Drawing inferences and conclusions about sample data and • Building useful decision models that incorporate uncertainty and risk Dr. C. Ertuna 34 Next Lesson (Lesson - 04/A) Statistical Sampling &Analysis of Sample Data Dr. C. Ertuna 35