MAT204_Lecture_04.pptx

# MAT204_Lecture_04.pptx - FACULTY OF INFORMATION TECHNOLOGY...

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Ta Quang Hung, Ph.D. FACULTY OF INFORMATION TECHNOLOGY Probability & Statistics Spring, 2017 Lecture 4: Random Variables & Distributions – Part 2

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Lecture 4 Probability & Statistics Contents Page 2 2 Bivariate Distributions 4 Conditional Distributions 3 Marginal Distributions 5 Summary 1 Cumulative Distributions
Lecture 4 Probability & Statistics 1. Definition & Properties of Cumulative Distribution Functions 2. c.d.f. of a Discrete & Continuous Distributions 3. Quantile Function Page 3 CUMULATIVE DISTRIBUTIONS

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Lecture 4 Probability & Statistics Definition of Cumulative Distribution Function The cumulative distribution function (c.d.f.) of a random variable is the function this function it can be used for both discrete and continuous random variables. It should be emphasized that the cumulative distribution function is defined as above for every random variable , regardless of whether the distribution of is discrete, continuous, or mixed. Page 4
Lecture 4 Probability & Statistics Example for c.d.f. of Discrete Variable Example 1: Machine breakdown Page 5 The probability mass function can be used to construct the cumulative distribution function as follows: Notice that the cumulative distribution function is a step function that starts at a value of 0 for small values of and increases to a value of 1 for large values of . The steps occur at the points and which are the possible values of the cost, and the sizes of the steps at these points and are simply the values of the probability mass function .

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Lecture 4 Probability & Statistics Example for c.d.f. of Discrete Variable Example 2: Binomial distribution Page 6 The p.d.f. of Binomial distribution The c.d.f. of Binomial distribution
Lecture 4 Probability & Statistics Example for c.d.f. of Continuous Variable Example 3: Metal Cylinder Production Suppose that the diameter of a mental cylinder has a probability density function Page 7 The probability density function for metal cylinder diameter This is a valid probability density function because The cumulative distribution function of the metal cylinder diameters can be constructed from the probability density function as

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Lecture 4 Probability & Statistics Example for c.d.f. of Continuous Variable Page 8 The c.d.f. is an increasing function between the limits and with ,
Lecture 4 Probability & Statistics Example for c.d.f. of Continuous Variable Example 4: Normal distribution Page 9 The p.d.f. of Normal distribution The c.d.f. of Normal distribution

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Lecture 4 Probability & Statistics
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• Spring '18
• Probability distribution, Probability theory, probability density function, New South Wales, Cumulative distribution function

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