distribution - Distribution and sampling Discrete random...

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1 Distribution and sampling Discrete random variable Continuous random variable Sampling and central limit theorem.
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2 Random Variables A random variable is a numerical description of the outcome of an experiment. A mapping from event to a real value. A discrete random variable may assume either a finite number of values or a infinite sequence of values. (usually count data) A continuous random variable may assume any numerical value in an interval or collection of intervals. (e.g. time, distance, weight etc.)
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3 Examples of Random Variables Experiment Our Interest Outcome (X) Flip 3 coins # of heads 2 Make 100 sales call # of sales 21 Manufacture a product Production time in minute 21.50
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4 Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. For a discrete random variable x , the probability distribution is defined by a probability function , denoted by f (x), which provides the probability for each value of the random variable. The required conditions for a discrete probability function are: f ( x ) > 0 f ( x ) = 1
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5 Expected Value and Variance The expected value , or mean , of a random variable is a measure of its central location. Expected value of a discrete random variable: E (x ) = = xf (x) The variance summarizes the variability in the values of a random variable. Variance of a discrete random variable: Var(x) = 2 = (x - ) 2 f (x) It can be interpreted as the descriptive measure of dispersion among the values of the random variable over a large number of repeats of the experiments. The standard deviation , , is defined as the positive square root of the variance.
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6 Example Expected Value of a Discrete Random Variable What is the expected number of TV sets sold in a day? x f ( x ) xf ( x ) 0 .40 .00 1 .25 .25 2 .20 .40 3 .05 .15 4 .10 .40 1.20= E ( x ) The expected number of TV sets sold in a day is 1.2
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7 The Binomial Probability Distribution Properties of a Binomial Experiment The experiment consists of a sequence of n identical trials. Two outcomes, success and failure , are possible on each trial. The probability of a success, denoted by p , does not change from trial to trial. The trials are independent . Our Interest is the number of successes occurring in n trials. Let x denotes the number of successes in n trials. x=0, 1, …, n
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8 Examples of Binomial Process (Bernoulli) In all cases we are interested in the probability of x successes in the n trials . Experiment with n trials Outcome per trial Probability of Success p Success Failure Flip a coin Head Tail 0.50 Inspect a part Good Defective 0.95 Contact a customer Sale No sale 0.20 Telephone survey Response No response 0.30
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9 Example: Evans Electronics Binomial Probability Distribution Evans is concerned about a low retention rate for employees. On the basis of past experience, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employees chosen at random, management estimates
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distribution - Distribution and sampling Discrete random...

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