# Note 2 - X is equally likely to be each of the integers a,....

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Math 151 – Probability Spring 2011 Jo Hardin Distributions random variable Let S be the sample space for an experiment. A real-valued function that is deﬁned on S is called a random variable. distribution Let X be a random variable. The distribution of X is the collection of all probabilities of the form P ( X C ) for all sets C of real numbers such that { X C } is an event. discrete A random variable X is discrete if it can take only a ﬁnite number k of diﬀer- ent values, x 1 ,x 2 ,...,x k or, at most, an inﬁnite sequence of diﬀerent values x 1 ,x 2 ,... . probability function If a random variable X has a discrete distribution, the proba- bility function of X is deﬁned as the function f such that for every real number x, f ( x ) = P ( X = x ) Bernoulli A random variable Z that takes only two values 0 and 1 with P ( Z = 1) = p has a Bernoulli distribution with parameter p . uniform Let a b be integers. Suppose that the value of a random variable
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Unformatted text preview: X is equally likely to be each of the integers a,. ..,b . Then we say that X has a uniform distribution on the integers [ a,b ]. • binomial A binomial random variable is a sum of Bernoulli random variables. • continuous A random variable X is continuous if there exists a nonnegative function f , deﬁned on the real line, such that for every interval of real numbers, the probability that X takes a value in the interval is the integral of f over the interval. For example: P ( a ≤ X ≤ b ) = Z b a f ( x ) dx • probability density function If X has a continuous distribution, the function f described above is called the probability density function (pdf). • cumulative distribution function The cumulative distribution function (cdf) of a random variable is the function: F ( x ) = P ( X ≤ x )...
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## This note was uploaded on 02/17/2012 for the course MATH 151 taught by Professor Jo.h during the Spring '10 term at Pomona College.

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