PROBABILITY, STATISTICS, AND RANDOM PROCESSES EE ?· PROBABILITY, STATISTICS, AND RANDOM PROCESSES EE…

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<ul><li><p>Tom Penick tom@tomzap.com www.teicontrols.com/notes ProbabilityStatisticsRandomProc.pdf 5/18/2001 Page 1 of 25 </p><p>PROBABILITY, STATISTICS, AND RANDOM PROCESSES EE 351K </p><p>INDEX ! factorial .................... 3, 24 1st fundamental theorem of probability...................... 16 </p><p>2nd fundamental theorem of probability...................... 17 </p><p>absorbing matrices............ 21 absorption </p><p>probability................... 21 time ............................ 21 </p><p>approximation theorem..... 18 area </p><p>sphere ......................... 25 Bayes' formula ................. 11 Bayes' inverse problem..... 11 Bayes' theorem................. 11 Bernoulli trials </p><p>central limit theorem.... 17 Bernoulli trials process ... 6, 7 beta density function......... 12 binomial coefficient............ 5 binomial distribution </p><p>central limit theorem.... 17 negative ........................ 7 </p><p>binomial distribution function........................................ 6 </p><p>binomial expansion........... 25 binomial theorem ............. 25 birthday problem.............. 23 B-matrix........................... 21 calculus............................ 24 canonical form ................. 21 card problem.................... 23 central limit theorem......... 17 </p><p>general form................ 18 sum of discrete variables</p><p>.............................. 18 central limit theorem for Bernoulli trials................ 17 </p><p>central limit theorem for binomial distributions ..... 17 </p><p>Chebyshev inequality ....... 16 coefficient </p><p>binomial........................ 5 multinomial................... 5 </p><p>coefficients of ordinary generating function ......... 20 </p><p>combinations...................... 3 conditional density function </p><p>continuous................... 11 conditional probability 10, 11 continuous conditional density function .............. 11 </p><p>continuous uniform density.9 convolution ...................... 15 </p><p>example ...................... 16 correlation........................ 15 covariance........................ 15 cumulative distribution function............................ 6 </p><p>cumulative normal distribution function.......... 6 </p><p>cyclic permutation .............. 3 </p><p>D(X) standard deviation ... 14 De Morgans laws ............ 10 density function.................. 8 </p><p>beta............................. 12 exponential.................... 9 joint .............................. 9 normal .......................... 8 standard normal............. 8 </p><p>dependent variable............ 25 derangement..................... 25 derivatives........................ 24 deviation .......................... 14 disoint................................ 3 distribution </p><p>hypergeometric.............. 7 uniform....................... 25 </p><p>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 </p><p>double coin toss.................. 2 e natural number.............. 24 E(X) expected value......... 12 envelope problem............. 23 ergodic chain.................... 22 Euler's equation................ 24 events </p><p>independent................... 4 ex infinite sum ................. 24 expectation </p><p>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 </p><p>f(x) density function........... 8 F(x,y) joint cumulative distribution function.......... 6 </p><p>f(x,y) joint density function 9 f() continuous uniform density function ................ 9 </p><p>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 </p><p>fX(x) normal density function........................................ 8 </p><p>g(t) generating function... 18, 19 </p><p>general math..................... 24 generating function..... 18, 19 </p><p>ordinary ...................... 19 properties .................... 20 </p><p>geometric distribution......... 7 glossary............................ 25 graphing terminology ....... 25 h(z) ordinary generating function.......................... 19 </p><p>hat check problem ............ 23 hypergeometric distribution 7 hypotheses ....................... 11 inclusion-exclusion principle</p><p>........................................ 3 independent events ............. 4 independent trials ............. 25 independent variable......... 25 inequality </p><p>Chebyshev .................. 16 Markov ....................... 13 </p><p>infinite sample space .......... 3 infinite sum...................... 24 integration........................ 24 intersection......................... 3 joint cumulative distribution function............................ 6 </p><p>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</p><p>...................................... 22 m() distribution function..5 Markov chains.................. 20 Markov inequality ............ 13 mean................................ 12 median ............................. 25 medical probabilities......... 23 memoryless property ........ 12 moment generating function</p><p>................................ 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 </p><p>N-matrix .......................... 21 normal density function ...... 8 normal random variable ...... 4 normalization ..................... 8 NR-matrix........................ 21 ordering ............................. 3 ordinary generating function</p><p>...................................... 19 </p><p>coefficients.................. 20 p(j) coefficients of the ordinary generating function...................................... 20 </p><p>permutation........................ 3 P-matrix........................... 21 poisson distribution ............ 7 poker probabilities............ 10 posterior probabilities ....... 11 prior probabilities ............. 11 probabilities </p><p>posterior...................... 11 prior............................ 11 </p><p>probability.......................... 2 uneven .......................... 4 </p><p>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 </p><p>normal .......................... 4 standard normal............. 4 </p><p>regular chain .................... 22 properties .................... 22 </p><p>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</p><p>...................................... 19 sphere .............................. 25 standard deviation .... 6, 8, 14 standard normal density function............................ 8 </p><p>standard normal random variable ............................ 4 </p><p>standardized sum.............. 17 states................................ 20 step .................................. 20 stirling approximation....... 24 stochastic ......................... 25 sum </p><p>standardized ................ 17 sum of continuous variables</p><p>...................................... 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 </p><p>probability................... 11 </p></li><li><p>Tom Penick tom@tomzap.com www.teicontrols.com/notes ProbabilityStatisticsRandomProc.pdf 5/18/2001 Page 2 of 25 </p><p>trials independent................. 25 </p><p>trigonometric identities..... 24 uneven probability.............. 4 uniform density .................. 9 uniform distribution...... 5, 25 union.................................. 3 V(X) variance .................. 14 variable </p><p>random........................ 25 </p><p>variance ........................... 14 continuous random </p><p>variables................. 14 discrete random variables</p><p>.............................. 14 properties .................... 14 </p><p>vector fixed probability.......... 22 </p><p>volume sphere ......................... 25 </p><p>W fixed probability matrix22 w limiting vector ............. 22 weak law of large numbers16 Z fundamental matrix of an ergodic chain.................. 22 </p><p>Z standard normal random variable ............................ 4 </p><p> sample space ............. 3, 5 (x) standard normal density function............................ 8 </p><p> failure rate...................... 9 expected value.............. 12 n moments ..................... 19 correlation.................... 15 standard deviation ........ 14 2 variance...................... 14 </p><p>PROBABILITY </p><p>Mathematically, the probability of an outcome is equal to the number of possible positive outcomes divided by the total possible outcomes (size of the sample space). </p><p>( ) number of possible positive outcomespositive outcometotal number of possible outcomes</p><p>P = </p><p>For example, if there are 5 balls in a box and 3 are green, the probability of choosing a green ball is </p><p>( ) 3choosing green ball5</p><p>P = </p><p>SINGLE COIN TOSS </p><p>There is an equal probability that the outcome will be heads or tails. </p><p>( ) ( ) 12</p><p>P H P T= = </p><p>DOUBLE COIN TOSS </p><p>There are four possible outcomes with equal probability. </p><p>( ) ( ) ( ) ( ) 14</p><p>P HH P HT P TH P TT= = = = </p><p>The probability of getting at least one heads would be the sum of the outcomes providing that result: </p><p>( ) ( ) ( ) 34</p><p>P HH P HT P TH+ + = </p><p>Alternatively, the probability of getting at least one heads could be thought of as one minus the probability of not getting at least one heads: </p><p>( ) 314</p><p>P TT = </p><p>MULTIPLE COIN TOSS There are 2n possible outcomes to a multiple coin toss (when considering order of the results). </p><p>The probability of getting at least one heads: </p><p>( ) ( ) 1getting at least one heads 2 12</p><p>nn</p><p>P = </p><p>( ) 1getting at least one heads 12n</p><p>P = </p><p>The probability of getting exactly 2 heads: </p><p>}( )</p><p>{ {</p><p>number ofnumberpossibleof possiblepositions positionsof secondof firstheadsheads</p><p>order of occurance inverseis not a factor outcome</p><p>1 1getting exactlytwo heads 2 2n</p><p>n nP</p><p> = </p><p>678 </p><p>The probability of getting exactly 3 heads: </p><p>}( )( ){</p><p>number of number ofnumberpossible possibleof possiblepositions positionspositionsof second of thirdof firstheads headsheads</p><p>number of ways toorder 3 items</p><p>1 2getting exactlythree heads 2 3</p><p>n n nP</p><p> = </p><p>678 678</p><p>{inverseoutcome</p><p>12n</p></li><li><p>Tom Penick tom@tomzap.com www.teicontrols.com/notes ProbabilityStatisticsRandomProc.pdf 5/18/2001 Page 3 of 25 </p><p>SET PROPERTIES </p><p>Given two sets A and B A B </p><p>The union of two sets refers to that which is in set A or set B. In terms of area, it is the sum of the areas minus the common area. </p><p>A B </p><p>The intersection of two sets refers to that which is in set A and set B. In terms of area, it is the common area found in both A and B. </p><p>A B </p><p>Two sets with no common elements are called disjoint. More than two such sets are called mutually exclusive. </p><p>A B </p><p>Theorems: ( ) ( ) ( ) ( )P A B P A P B P A B = + </p><p> ( ) ( ) 1P A P A+ = A means everything that is not in A </p><p>INCLUSION-EXCLUSION PRINCIPLE </p><p>Expanding on a theorem presented in the previous box, the probability that at least one event contained within a group of (possibly) intersecting sets of events occurs is the sum of the probabilities of each event minus the sum of the probabilities of all 2-way intersections, plus the sum of the probabilities of all 3-way intersections, minus the sum of all 4-way intersections, etc. </p><p>( ) ( ) ( )</p><p>( )</p><p>1 21 1</p><p>1</p><p>n</p><p>n i i ji i j n</p><p>i j ki j k n</p><p>P E E E P E P E E</p><p>P E E E</p><p>= &lt; </p><p> &lt; &lt; </p><p> = </p><p>+ </p><p>L</p><p>L </p><p>E = an event, in this case </p><p> SAMPLE SPACE The set of all possible outcomes. (Means the same as probability space, I think.) For example, if a coin is tossed twice, the sample space is </p><p>( ) ( ) ( ) ( ){ }H, H , H, T , T, H , T, T = Note that the sample space is not a number; it is a collection or set of results. This is frequently a source of confusion; the size of the sample space is a number, but the sample space itself is not a number. See Probability, p2. </p><p> INFINITE SAMPLE SPACE If a coin is tossed until it turns up heads, then the sample space for possible outcomes is </p><p>{ }1, 2, 3, = </p><p>PERMUTATION </p><p>A permutation is a mapping of a finite set onto itself. For example if we have the set A = {a,b,c}, there are 3! possible permutations. One of them is </p><p>a b c</p><p>c b a </p><p>In other words, there are n! ways to arrange n objects. However, in the case of a cyclic permutation, there are (n-1)! ways to arrange n objects. An example of a cyclic permutation would be the seating of n people at a round table. </p><p>ORDERING/COMBINATIONS 1 </p><p>n different objects can be ordered in n! ways or permutations. What about the case when some objects are identical? For example, consider the letters in the name </p><p>KONSTANTOPOULOS There are 15 letters in the word. This includes 4 Os, 2 Ns, 2 Ts, and 2 Ss. How many 15-letter combinations can be formed with these letters given that some letters are identical? </p><p>15!number of words</p><p>4!2!2!2!= </p><p>ORDERING/COMBINATIONS 2 </p><p>In how many ways can n identical objects be arranged in m containers? </p><p>( )( ) ( )</p><p>1 1 !</p><p>1 1 ! !</p><p>n m n m</p><p>n n m n</p><p>+ + = </p><p>A problem that appeared in the textbook was, how may ways can 6 identical letters be put in 3 mail boxes? The answer is 28. </p></li><li><p>Tom Penick tom@tomzap.com www.teicontrols.com/notes ProbabilityStatisticsRandomProc.pdf 5/18/2001 Page 4 of 25 </p><p>NORMAL RANDOM VARIABLE </p><p>A normal random variable has a Gaussian density function centered at the expectation . The figure below shows plots of the density functions of two normal random variables, centered at the common expectation of 0. The plot having the sharper peak is for the special case of a standard normal random variable determined by and expectation of 0 and deviation of 1. A normal random variable does not necessarily have an expectation of zero...</p></li></ul>

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