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**Review of Basic Probability and Statistics**

ISE525: Spring 10

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**Random Variables and Their Properties**

Experiment : a process whose outcome is not known with certainty. Set of all possible outcomes of an experiment is the sample space. Outcomes are sample points in the sample space. The distribution function (or the cumulative distribution function, F(x), of the random variable X is defined for each real number as follows:

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**Properties of distribution functions**

1) 2) F(x) is nondecreasing. 3)

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**Discrete Random Variables**

A random variable, X, is said to be discrete if it can take on at most a countable number of values: The probability that X takes on the value xi is given by Also: p(x) is the probability mass function.

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**Discrete variables continued:**

The distribution function F(x) for the discrete random variable X is given by:

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Moments

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**Moments of a Probability Distribution**

The variance is defined as the average value of the quantity : (distance from mean)2 The standard deviation, σ =

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**For discrete Random Variables**

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**Continuous random variables**

A random variable X is said to be continuous if there exists a non-negative function, f(x), such that for any set of real numbers B, Unlike a mass function, for the continuous random variable, f(x) is not the probability that the random number equals x.

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**Multiple random variables**

IF X and Y are discrete random variables, then the joint probability mass function is: P(x,y) = P(X=x, Y=y) X and Y are independent if:

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**Multiple random variables**

For continuous random variables, the joint pdf is For independence:

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Properties of means This holds even if the variables are dependent!

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**Properties of variance**

This does not hold if the variables are correlated.

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**Common Discrete Distributions**

Bernoulli: Coin toss Binomial: Sum of Bernoulli trials

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**Poisson Distribution:**

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**Common Continuous Distributions**

Uniform

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**Exponential Distribution**

Probability distribution function (pdf) and the Cumulative distribution functions (cdf) are: Mean and Standard Deviation are:

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**Common Continuous Distributions**

Normal Distribution:

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Other Distributions Erlang distribution:

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Gamma Distribution

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**Estimation Means, variances and correlations:**

Simulation data are almost always correlated (according to Law and Kelton) !

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**Hypothesis tests for means**

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**Strong law of large numbers**

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