Lesson10S1.pptx - TAIBAH UNIVERSITY Faculty of...

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TAIBAH UNIVERSITY Faculty of Science Department of Math . ةبيط ةعماج مولعلا ةيلك تايضايرلا مسق Probability and Statistics for Engineers STAT 305 Teacher : Dr.Moustapha Abdellahi First Semester 1438/1439
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Random Variables and Probability Distributions Lesson 10
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Concept of Random Variable Contents Discrete Probability Distribution and Cumulative Distribution Function Continuous Probability Distribution and Cumulative Distribution Function
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In a statistical experiment, it is often very important to allocate numerical values to the outcomes. Experiment : testing two components. ( D =defective, N =non-defective) Sample space : S ={DD,DN,ND,NN} Concept of Random Variable Example
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Let X = number of defective components when two components are tested. Assigned numerical values to the outcomes are: Concept of Random Variable Sample point (Outcome) Assigned Numerical Value (x) DD 2 DN 1 ND 1 NN 0 Notice that, the set of all possible values of the random variable X is {0, 1, 2}.
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A random variable X is a function that associates each element in the sample space with a real number (i.e., X : S R.) " X “ ( capital letter ) denotes the random variable . " x " ( small letter ) denotes a value of the random variable X . Concept of Random Variable Definition : Notation
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Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. The possible outcomes and the values y of the random variable: Y , where Y is the number of red balls, are Example :
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Let X be the random variable defined by the: waiting time, in hours, between successive speeders spotted by a radar unit. The random variable X takes on all values x for which x > 0 . Example :
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A random variable X is called a discrete random variable if its set of possible values is countable, i.e., x { x 1 , x 2 , …, x n } or x { x 1 , x 2 , …} In most practical problems: A discrete random variable represents count Types of Random Variable Discrete Random Variable :
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A random variable X is called a continuous random variable if it can take values on a continuous scale, i.e., x {x: a < x < b; a, b R} In most practical problems: A continuous random variable Types of Random Variable Continuous Random Variable:
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A discrete random variable X assumes each of its values with a certain probability. Example: Experiment : tossing a non- balance coin 2 times independently. H= head , T=tail Sample space: S ={HH, HT, TH, Probability Distributions (Discrete ) Discrete Probability Distributions
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Sample point (Outcome) Probability Value of X (x) HH P(HH)=P(H) P(H)=1/3 1/3 = 1/9 2 HT P(HT)=P(H) P(T)=1/3 2/3 = 2/9 1 TH P(TH)=P(T) P(H)=2/3 1/3 = 2/9 1 TT P(TT)=P(T) P(T)=2/3 2/3 = 4/9 0 Probability Distributions (Discrete ) Suppose P(H) = ½P(T) P(H)= 1/3 and P(T)= 2/3 Let X = number of heads
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