Probability Theory Random Variables Random Processes

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1 Foundametnal Course on robability, Random Variable and Random rocesses robability Theory Random Variables Random rocesses Teacher: W.K. Cham tationary R & Ergodic R Gaussian R Filtering of R

2 robability Theory - 3 stages In order to develop a useful theory of probability, it is important to separate 3 stages in the consideration of any real probability.. The association of an events with a probability by i experiments and ii reasoning. e.g. = /6 2. Development of the relationship of the probability of an event with the probabilities of some other events. e.g. 3. The application of the results of stage & stage 2 to the real world. e.g. 2

3 robability Theory - some definitions { discrete * continuous single performance is called a trial & at it we observe a single outcome i. The event is said to have occurred in this trial if i. e.g. outcome i = event = {,2,3} occurred. robability space = universal set = e.g. = {,2,,6} et contains all possible experimental outcomes φ = empty set = set contains impossible outcomes 3

4 tage : The association of an events with a probability For example, we are to determine the probability of event which is the outcome being one in a trial of throwing a dice. robability determined by experiment The experiment of throwing a dice is repeated n times. uppose n a times of the n trials result in event. robability of = Comments: i not exact! lim n ii n n a robability determined by reasoning n a n provided n is sufficiently large. may be exact but cannot be found in practice. We may assume that throwing a dice has six possible outcomes and so there are 6 possible events. If all events have the same probability, then = /6. Comment: not exact as the assumptions may be wrong. 4

5 tage 2: Development of the relationship between probabilities of different events a trial pace outcome = Event {} occurred. {,2,,6} We assign to each event a number which we call the probability of. This number satisfies the 3 axioms: = 3. = φ + = + i.e. mutually exclusive Th m.. = φ = = + + = + 5

6 Fill in the missing words in Definitions:- Event universal set occurs at trial. φ 2 Event empty set occurs. 3 Event + occurs when event event occur. 4 Event occurs when event event occur. roperties:- =0 event & event occur in the same trial. 2 Event occurs Event occurs. 6

7 Two Theorems in robability Theory rove: roof: = =φ set theory + set theory = + = + = = axiom 3 Q.E.D. rove: φ = 0 roof: = + = +φ =φ + φ = + φ = φ = = 0 set theory set theory Q.E.D. 7

8 Three definitions in robability Theory Given event & event, we have The probability of the occurrence of events &. + / The probability of the occurrence of event or. The probability of the occurrence of event given. In general, we have CD. & + / ++C+D+. 8

9 robability Theory Def. The conditional event for given, /, is the event under the stipulation that has occurred. Def. The conditional probability of given is / /. roperties 2 3 = 0 / = = = 4 = {,2,3} = {,3,5} 0 / = / = = / 3 / = = = / Examples

10 robability Theory Def. Two events & are independent if. Two events are independent when knowledge of the occurrence of one event gives no additional information concerning the likelihood of the occurrence of a second event. / = = = e.g. trial trial 2 = {,2,3,4,5,6} ={} = 6 = {,2,3,4,5,6} 2 ={} 2 = 6 2 = 2 if & 2 are independent. The space of 2 is = x 2 = {,,,2,,3,, 6,6}. Event 2 consisting of all ordered-pairs i, 2j s.t. i 2j 2 is a subset of.

11 Mutually Exclusive Events and Independent Events e.g. = {} = {2} φ / = = = 0 a single trial & are mutually exclusive, i.e. =0 e.g. 2 = {} = {2} trial trial 2 / = = = & are independent, i.e. =

12 Independent Events Def. Events, 2 & 3 are independent iff 2 3 = 2 3 & 2 = 2 & 3 = 3 & 2 3 = 2 3. Def. In general, n events, 2,, n are independent iff 2 n = 2 n : : i j k = i j k i j = i j for all combinations of i, j, k, where i j k n. Example: Given: = /2 2 = /4 3 = /4 2 = /8 3 = /8 2 3 = /8 2 3 = /32 re, 2 & 3 independent? 2

13 robability Theory Th m. For events, 2,, i which may or may not be independent, the probability of the simultaneous occurrence of the i events is 2 i = 2 / 3 / 2 i / 2 i-. Example: Let, 2, 3 & 4 represent the consecutive events of drawing an aces. Find: olution: = 2 / 3 / 2 4 / = =

14 4 ROILITY THEORY set theory Defintions: trial outcome event 3 axioms φ + {}... + = + = = single trial Theorem of Total robability Definition: Conditional robability Multiplicative Law / = / n = /... / n n + + = multiple trials = ] ] ] new space = X 2 space space 2 Definition: Events & are independent

Axioms of Probability. Set Theory. M. Bremer. Math Spring 2018

Axioms of Probability. Set Theory. M. Bremer. Math Spring 2018 Math 163 - pring 2018 Axioms of Probability Definition: The set of all possible outcomes of an experiment is called the sample space. The possible outcomes themselves are called elementary events. Any