# Topic1 - Topic 1 Probability foundations Brendan K Beare 1...

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Topic 1: Probability foundations Brendan K. Beare September 22, 2011 1 Mass, density and distribution functions A discrete random variable X can be equal to any of a finite number of possible values. The probabilities with which X takes on each of those possible values are given by its probability mass function , or pmf, which we denote f X . The probability that X is equal to some fixed value x is equal to f X ( x ): P ( X = x ) = f X ( x ) . Since the values of f X ( x ) for different x ’s represent probabilities, we must have f X ( x ) 0 for all x , and the sum of the probabilities must equal one: x f X ( x ) = 1. A continuous random variable X is equal to any fixed value x with probability zero. The random behavior of X is described by its probability density function , or pdf, which we also denote f X . Probability density and mass functions are not the same thing. When X is a continuous random variable, the probability that X lies in an interval ( a, b ) is equal to the area under its pdf between a and b : P ( a < X < b ) = Z b a f X ( x )d x. Since the probability of a random variable lying in any interval ( a, b ) must be nonnegative, every pdf must be nonnegative: f X ( x ) 0 for all x . Also, the probability of a random variable lying anywhere on the real line is equal to one, so we must have R -∞ f X ( x )d x = 1 We can describe the behavior of discrete or continuous random variables using their cumulative distribution function , or cdf, which we denote F X . The probability that X is equal to or less than some fixed value x is equal to F X ( x ): P ( X x ) = F X ( x ) . Every cdf is an nondecreasing function that increases from zero to one as we move rightward along the x -axis. When X is discrete, the probability that it is equal to 1

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or less than x is equal to the sum of its pmf f X over all values of the argument that are equal to or less than x : F X ( x ) = X y x f X ( y ) . When X is continuous, the probability that it is equal to or less than x is equal to the area under its pdf f X to the left of x : F X ( x ) = Z x -∞ f X ( y )d y. If F X is differentiable, X must be a continuous random variable, and its pdf f X is the derivative of F X . If F X is a step function, then X is a discrete random variable, and its pmf f X ( x ) is equal to zero if F X is continuous at x , and equal to the size of the discontinuity of F X at x for those x where F X is discontinuous. 2 Expected value and variance The expected value of a discrete random variable X with pmf f X is defined as E ( X ) = X x xf X ( x ) . More generally, the expected value of g ( X ) for some real valued function g is defined as E ( g ( X )) = X x g ( x ) f X ( x ) . Similarly, the expected value of a continuous random variable X with pdf f X is defined as E ( X ) = Z -∞ xf X ( x )d x, and the expected value of g ( X ) is defined as E ( g ( X )) = Z -∞ g ( x ) f X ( x )d x.
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