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Unformatted text preview: Chapter 4 summary Random Variables Definition of random variable: A random variable is a function that assigns a real number to the result of a random experiment. Thus it is a function whose domain is the sample space of the experiment and whose range is contained in the set of real numbers. Discrete and continuous random variables: The random variable is called discrete if its range is either finite (for example { 1 , 5 , 7 } ) or countably infinite (for example { , 1 , 2 , 3 , 4 , 5 ,... } . We wait until Chapter 5 to define what are continuous random variables. In this chapter we consider only discrete random variables. Probability mass function of a discrete random variable: If X is a discrete random variable, its probability mass function is the function p ( x ) = P ( X = x ) . It is necessarily 0 and x p ( x ) = 1, where the sum is taken over all possible values of X . Thus p is a probability measure on the range of X . Distribution of a random variable: If X is any random variable (continuous or discrete), the distribution function of X is the function F defined for any real number x by F ( x ) = P ( X x ) . It is also referred to as the cumulative distribution or cdf of X . It always has the following three properties: (i) lim x  F ( x ) = 0, (ii) lim x F ( x ) = 1, and (iii) F is right continuous. If X is discrete, then F is a step function, i.e. its graph consists of horizontal straight lines. Calculating the cdf from the probability mass function and the probability mass function from the cdf: We have F ( x ) = X t x P ( X = t ) . Conversely, if the range of X is finite, and x 1 < x 2 are two consecutive range values of X , then P ( X = x 2 ) = F ( x 2 ) F ( x 1 ) . Calculating probabilities using the distribution: If we know the distribution of the discrete random variable X , then various probabilities of interest can be calculated directly from F as follows: P ( X x ) = F ( x ) P ( X = x )...
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 Spring '08
 Moumen,F
 Probability

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