Review of Basic Probability and Statistics ISE525: Spring 10.
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Transcript of Review of Basic Probability and Statistics ISE525: Spring 10.
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:
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.
Discrete variables continued:
• The distribution function F(x) for the discrete random variable X is given by:
Moments of a Probability Distribution
• The variance is defined as the average value of the quantity : (distance from mean)2
• The standard deviation, σ =
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.
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:
Exponential Distribution
• Probability distribution function (pdf) and the Cumulative distribution functions (cdf) are:
• Mean and Standard Deviation are:
Estimation
• Means, variances and correlations:
• Simulation data are almost always correlated (according to Law and Kelton) !