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W3INSE6220

# W3INSE6220 - 1 3 Histogram Useful for large data sets INSE...

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1 INSE 6220 -- Week 3 Advanced Statistical Approaches to Quality Descriptive Statistics Discrete Probability Distributions Continuous Probability Distributions Dr. A. Ben Hamza Concordia University 2 Random Variables and Probability Density Functions A random variable is a quantity whose value is not known exactly but its probability distribution is known. The value of the random variable will vary from trial to trial as the experiment is repeated. The variable’s probability density function ( PDF ) describes how these values are distributed (i.e. it gives the probability that the variable value falls within a particular interval). Smallest values are most likely y f y ( y ) Exponential distribution (e.g. event rainfall) 0 0 3 1 2 4 y f y ( y ) Discrete distribution (e.g. number of severe storms) Only discrete values (integers) are possible Probability that y = 2 0.2 0.3 0.25 0.15 0.1 0 1 All values between 0 and 1 are equally likely y f y ( y ) Uniform distribution (e.g. soil texture) Continuous PDFs A Discrete PDF 3 Histogram - Useful for large data sets • Group values of the variable into bins, then count the number of observations that fall into each bin • Plot frequency (or relative frequency) versus the values of the variable 4 Sometimes called a probability mass function Sometimes called a probability density function

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5 Mean of a Random Variable 6 Variance of a Random Variable ( ) is referred to as the standard deviation Var X The (population) variance of random variable (RV) gives an idea of how widely spread the values of the RV are likely to be. It is the second moment of the distribution, indicating how closely concentrated around the expected value of the distribution is. The variance is defined by 2 2 2 ( ) ( ) ( ( )) Var X E X E X Example 7 Sample mean and sample variance Mean variance 2 s s 8 Bernoulli Trial: · Bernoulli trial is an experiment with only two possible outcomes. · The two possible outcomes are labeled: success ( s ) and failure ( f ) · The probability of success is P( s )= p and the probability of failure is P( f )= q = 1 p . Examples: 1. Tossing a coin (success=H, failure=T, and p =P(H)) 2. Inspecting an item (success=defective, failure=non-defective, and p =P(defective)) Binomial Distribution
9 Binomial Probability Distribution Our interest is in the number of successes occurring in the n trials. We let x denote the number of successes occurring in the n trials. Number of Experimental Outcomes Providing Exactly x Successes in n Trials where: n ! = n ( n – 1)( n – 2) . . . (2)(1) 0! = 1 ! !( )! n n x x n x       10 The probability distribution of X is given by: (1 ) ; 0,1, 2, , ( ) ( ) ( ; , ) 0 ; x n x n p p x n f x P X x b x n p x otherwise       f ( x ) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial

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W3INSE6220 - 1 3 Histogram Useful for large data sets INSE...

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