A First Course in ProbabilityFor upper level or undergraduate/graduate level introduction to probability for math, science, engineering, and business students with a background in elementary calculus. This market-leading introduction to probability features exceptionally clear explanations of the mathematics of probability theory and explores its many diverse applications through numerous interesting and motivational examples. The outstanding problem sets and intuitive explanations are hallmark features of this market leading text. |
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Page 13
... nonnegative ( as opposed to positive ) solutions , note that the number of nonnegative solutions of x1 + x2 + ··· + x , = n is the same as the number of positive solutions of y + + y , = n + r ( seen by letting y ; = x ; + 1 , i = 1 ...
... nonnegative ( as opposed to positive ) solutions , note that the number of nonnegative solutions of x1 + x2 + ··· + x , = n is the same as the number of positive solutions of y + + y , = n + r ( seen by letting y ; = x ; + 1 , i = 1 ...
Page 225
... nonnegative . When y = g ( x ) for any x , then Fy ( y ) is either 0 or 1 , and in either case fy ( y ) = 0 . Example 7d . Let X be a continuous nonnegative random variable with density func- tion f , and let Y = X " . Find fy , the ...
... nonnegative . When y = g ( x ) for any x , then Fy ( y ) is either 0 or 1 , and in either case fy ( y ) = 0 . Example 7d . Let X be a continuous nonnegative random variable with density func- tion f , and let Y = X " . Find fy , the ...
Page 320
... nonnegative random variables ( that is , P { X ; ≥ 0 } ∞ 2. Σ E [ | X ; | ] < ∞ . i = 1 = 1 for all i ) . Example 2q . Consider any nonnegative , integer - valued random variable X. If for each i≥ 1 , we define then 1 if X ≥ i X ...
... nonnegative random variables ( that is , P { X ; ≥ 0 } ∞ 2. Σ E [ | X ; | ] < ∞ . i = 1 = 1 for all i ) . Example 2q . Consider any nonnegative , integer - valued random variable X. If for each i≥ 1 , we define then 1 if X ≥ i X ...
Contents
AXIOMS OF PROBABILITY | 24 |
CONDITIONAL PROBABILITY AND INDEPENDENCE | 64 |
RANDOM VARIABLES | 122 |
Copyright | |
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approximately assume binomial random variable black balls cards central limit theorem Chebyshev's inequality components compute conditional probability Consider continuous random variable Cov(X defined denote the event denote the number desired probability dice discrete random variable distribution function distribution with parameters equal Equation Example expected number expected value exponential random variable Find the probability flips follows function F Hence HINT independent trials inequality joint density function large numbers least Let X denote moment generating function nonnegative normal random variable normally distributed obtain occur P(EF P₁ P₂ pair percent permutation player Poisson random variable preceding prob probability density function probability mass function problem Proposition random number random vari randomly chosen result sample space sequence Show simulate Solution Let Suppose uniformly distributed Var(X variable with mean variable with parameters white balls wins X₁ Y₁ Y₂ Σ Σ