Week 3 - Probability Distributions

# Week 3 - Probability Distributions - Engr9397 Week3...

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Engr 9397 – Week 3 Probability Distributions

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Probability Models Probability estimates can be obtained either through experimentation or by a theoretical model Example: number of times tails appears when a coin is tossed 3 times. We could conduct and experiment where we would toss a 3 coins 1000 times and observe the results are many tests available OR We could assume coins have an unbiased probability of 0.5 and reason probability would be P(X=3) = p 3 = 0.5 3 = 0.125
Types of Probability Distributions There are over 500 probability distributions! Probability distributions most commonly used in engineering applications include: Discrete Models Bernoulli sequence, Binomial, Poisson Continuous Models Uniform, Normal, Lognormal, Exponential

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Probability Distribution Models Models are conceptual and are defined parameters (usually 3 or less) using estimates from the data We are often using point estimates only, which are subject to variability, so we need to check to see whether the model fits the data To check how well a given model matches with the data, goodness of fit tests are used
Probability distribution function conditions All probability models follow several rules In order for any function, f(x) to represent a probability distribution, the following conditions must be satisfied: Each value of f(x) must lie between 0 and 1, inclusive (has nonnegative densities or masses) The area under the probability distribution must equal 1 (sum of all values of f(x) must equal 1) Only the equation and shape of the distribution curve is different, and the choice of appropriate distribution depends on the data

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Discrete Probability Distributions Probability Distribution Function arguments (x) are values of a random variable functional values f(x) are probabilities Mean: = Variance: 2 = Recall that a random variable can be either continuous or discrete x123456 F(x) .15 .25 .3 .15 .1 .05 allx x xf ) ( ) ( ) ( 2 x f x allx  
Bernoulli sequence and the Binomial distribution Many problems in engineering can be modeled as just 2 possible outcomes in a trial. Ex. yes or no, success or failure, head or tail Problems of this type may be modeled by a Bernoulli sequence The probability of x occurrences in n trials of a Bernoulli sequence is given by the Binomial distribution, a discrete probability distribution that has two parameters, n (# of trials) and p (probability of success)

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Binomial Distribution Binomial distributions are used when: Experiment consists of n trials/repetitions Each trial has two mutually exclusive outcomes The probability of occurrence (p) or non occurrence (q=1 p) of the event in each trial is constant Trials are independent of each other
Use of Binomial model In spite of its simplicity, the Binomial model is quite useful in many engineering applications It will work as long as we can determine and p, and use/assume individual, discrete trials For space

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Week 3 - Probability Distributions - Engr9397 Week3...

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