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| Summary |
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 P {Occurrence of A given that B has occurred} |
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| = P (A|B) |
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| Total probability: |
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| Baye's theorem: |
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| Let S be the simple space and E1, E2, E3,….,En be n mutually exclusive and exhaustive events associated with a random experiment. If A is any arbitrary event then |
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| Random Variable: random variable is a function which associates a real number to each outcome of a random experiment. |
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| Discrete random variable: Random variables which can assume only a finite number, or a countable infinity number of values, are called discrete random variables. |
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| Continuous random variable: random variables which can assume any value over an interval. |
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| Probability and Distribution |
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| If x is a discrete random variable assuming the values x1, x2, x3,….,xn with probabilities p1, p2, p3,…., pn respectively then (x1,p1), (x2, p2),…(xn, pn) defines a probability distribution of X. |
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| Mathematical Expectation of X or Mean of X is |
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| E (X) = x1p1 + x2p2 + ….+ xnpn |
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| Variance of X: If X is a discrete random variable then variance of X denoted by V(X) is defined as |
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| V(X) = E [X - E(X)]2 |
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| V(X) = E(X2) - [E(X)]2 |
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| Bernoulli trial: A random experiment which has only two outcomes (success or failure) |
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| Let n independent bernoulli trials be performed and X denote the number of successes in n trials then X follows a binomial distribution. |
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| The probability of success be p, |
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| The probability of failure = 1- p = q |
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| The probability of r success in n trials |
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| = P {X = r} = nCr pr qn-r is called the probability mass function of the binomial distribution. |
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| Mean of binomial distribution = E(X) = np |
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| Variance of binomial distribution = V(X) = npq |
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| Poisson distribution is a limiting case of binomial distribution when n is very large, p is very small and np is a constant denoted (the parameter) |
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Probability mass function of a poisson distribution is P(X = x)
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| Mean of poisson distribution E(X) = l |
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| Variance of poisson distribution V(X) = l |
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