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  • Tom Penick tom@tomzap.com www.teicontrols.com/notes ProbabilityStatisticsRandomProc.pdf 5/18/2001 Page 1 of 25

    PROBABILITY, STATISTICS, AND RANDOM PROCESSES EE 351K

    INDEX ! factorial .................... 3, 24 1st fundamental theorem of probability...................... 16

    2nd fundamental theorem of probability...................... 17

    absorbing matrices............ 21 absorption

    probability................... 21 time ............................ 21

    approximation theorem..... 18 area

    sphere ......................... 25 Bayes' formula ................. 11 Bayes' inverse problem..... 11 Bayes' theorem................. 11 Bernoulli trials

    central limit theorem.... 17 Bernoulli trials process ... 6, 7 beta density function......... 12 binomial coefficient............ 5 binomial distribution

    central limit theorem.... 17 negative ........................ 7

    binomial distribution function........................................ 6

    binomial expansion........... 25 binomial theorem ............. 25 birthday problem.............. 23 B-matrix........................... 21 calculus............................ 24 canonical form ................. 21 card problem.................... 23 central limit theorem......... 17

    general form................ 18 sum of discrete variables

    .............................. 18 central limit theorem for Bernoulli trials................ 17

    central limit theorem for binomial distributions ..... 17

    Chebyshev inequality ....... 16 coefficient

    binomial........................ 5 multinomial................... 5

    coefficients of ordinary generating function ......... 20

    combinations...................... 3 conditional density function

    continuous................... 11 conditional probability 10, 11 continuous conditional density function .............. 11

    continuous uniform density.9 convolution ...................... 15

    example ...................... 16 correlation........................ 15 covariance........................ 15 cumulative distribution function............................ 6

    cumulative normal distribution function.......... 6

    cyclic permutation .............. 3

    D(X) standard deviation ... 14 De Morgans laws ............ 10 density function.................. 8

    beta............................. 12 exponential.................... 9 joint .............................. 9 normal .......................... 8 standard normal............. 8

    dependent variable............ 25 derangement..................... 25 derivatives........................ 24 deviation .......................... 14 disoint................................ 3 distribution

    hypergeometric.............. 7 uniform....................... 25

    distribution function . 5, 6, 25 binomial........................ 6 cumlative ...................... 6 discrete uniform ............ 5 exponential.................... 8 geometric ...................... 7 joint .............................. 7 joint cumulative............. 6 multinomial................... 7 negative binomial .......... 7 poisson.......................... 7

    double coin toss.................. 2 e natural number.............. 24 E(X) expected value......... 12 envelope problem............. 23 ergodic chain.................... 22 Euler's equation................ 24 events

    independent................... 4 ex infinite sum ................. 24 expectation

    properties .................... 13 expectation of a function... 13 expectation of a product.... 13 expected value.......... 6, 8, 12 exponential density function9 exponential distribution ...... 8 f(t) exponential density function............................ 9

    f(x) density function........... 8 F(x,y) joint cumulative distribution function.......... 6

    f(x,y) joint density function 9 f() continuous uniform density function ................ 9

    factorial........................ 3, 24 failure rate.......................... 9 fixed probability matrix .... 22 fixed probability vector..... 22 fundamental matrix..... 21, 22 FX(x) cumulative normal distribution function.......... 6

    fX(x) normal density function........................................ 8

    g(t) generating function... 18, 19

    general math..................... 24 generating function..... 18, 19

    ordinary ...................... 19 properties .................... 20

    geometric distribution......... 7 glossary............................ 25 graphing terminology ....... 25 h(z) ordinary generating function.......................... 19

    hat check problem ............ 23 hypergeometric distribution 7 hypotheses ....................... 11 inclusion-exclusion principle

    ........................................ 3 independent events ............. 4 independent trials ............. 25 independent variable......... 25 inequality

    Chebyshev .................. 16 Markov ....................... 13

    infinite sample space .......... 3 infinite sum...................... 24 integration........................ 24 intersection......................... 3 joint cumulative distribution function............................ 6

    joint density function .......... 9 joint distribution................. 7 LHpitols rule................ 24 law of averages................. 16 law of large numbers ........ 16 linearizing an equation...... 25 logarithms ........................ 24 M mean first passage matrix

    ...................................... 22 m() distribution function..5 Markov chains.................. 20 Markov inequality ............ 13 mean................................ 12 median ............................. 25 medical probabilities......... 23 memoryless property ........ 12 moment generating function

    ................................ 18, 19 moments .......................... 19 multinomial coefficient....... 5 multinomial distribution ..... 7 multiple toss....................... 2 mutually exclusive.............. 3 NA(0,z) table of values..... 17 natural number e............... 24 Nc-matrix......................... 21 negative binomial distribution ....................... 7

    N-matrix .......................... 21 normal density function ...... 8 normal random variable ...... 4 normalization ..................... 8 NR-matrix........................ 21 ordering ............................. 3 ordinary generating function

    ...................................... 19

    coefficients.................. 20 p(j) coefficients of the ordinary generating function...................................... 20

    permutation........................ 3 P-matrix........................... 21 poisson distribution ............ 7 poker probabilities............ 10 posterior probabilities ....... 11 prior probabilities ............. 11 probabilities

    posterior...................... 11 prior............................ 11

    probability.......................... 2 uneven .......................... 4

    probability of absorption... 21 probability space ................ 3 probability tree ................. 11 problems .......................... 23 properties of expectation... 13 Q-matrix .......................... 21 quadratic equation ............ 25 random variable................ 25

    normal .......................... 4 standard normal............. 4

    regular chain .................... 22 properties .................... 22

    reliability ......................... 10 reverse tree....................... 11 R-matrix........................... 21 round table ......................... 3 sample problems............... 23 sample space .................. 3, 5 series................................ 25 set properties...................... 3 single coin toss................... 2 Sn* standardized sum....... 17 specific generating functions

    ...................................... 19 sphere .............................. 25 standard deviation .... 6, 8, 14 standard normal density function............................ 8

    standard normal random variable ............................ 4

    standardized sum.............. 17 states................................ 20 step .................................. 20 stirling approximation....... 24 stochastic ......................... 25 sum

    standardized ................ 17 sum of continuous variables

    ...................................... 15 sum of discrete variables... 15 table of values for NA(0,z) 17 time to absorption............. 21 t-matrix............................ 21 transient state ................... 21 transition matrix ............... 20 tree

    probability................... 11

  • Tom Penick tom@tomzap.com www.teicontrols.com/notes ProbabilityStatisticsRandomProc.pdf 5/18/2001 Page 2 of 25

    trials independent................. 25

    trigonometric identities..... 24 uneven probability.............. 4 uniform density .................. 9 uniform distribution...... 5, 25 union.................................. 3 V(X) variance .................. 14 variable

    random........................ 25

    variance ........................... 14 continuous random

    variables................. 14 discrete random variables

    .............................. 14 properties .................... 14

    vector fixed probability.......... 22

    volume sphere ......................... 25

    W fixed probability matrix22 w limiting vector ............. 22 weak law of large numbers16 Z fundamental matrix of an ergodic chain.................. 22

    Z standard normal random variable ............................ 4

    sample space ............. 3, 5 (x) standard normal density function............................ 8

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