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Detailed informtion about Probability, and Statistics , Introduction to Probability Theory, Queueing and Computer Science applications, Schaum’s Outline Series, Experiment, Deterministic Experiments.
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Probability, and Statistics Paul Mayer: Introduction to Probability Theory K.S. Trivedi: Introduction to Probability & Statistics with Reliability, Queueing and Computer Science applications. Theory and Problems of Statistics: Schaum’s Outline Series
What is Probability? What is an event? For describing what kind of experiments we use probability theory?
A precisely deterministic input yields a precisely deterministic output. We can predict the output of the experiment. Ex: Distance covered by a car traveling at a constant speed; Ohm's law; determining gravitational constant at a place etc.
Even exact knowledge of input and action does not allow exact prediction of outcome. Ex: Tossing a coin; throwing a dice; Life of an electric bulb; Number of road accidents in a day at Allahabad; Queue size at a railway reservation counter; etc. In probability theory we are mainly concerned with the random experiments.
Examples: Tossing a coin; ={H,T}H,T} Throw a dice; ={H,T}1,2,3,4,5,6} Toss a coin until we obtain a Head; ={H,T}H,TH,TTH,TTTH, …}; (countably }; (countably countably infinite) Life of a bulb (countably in hrs); ={H,T}t: 0t<} (countably Sample space has uncountable number of points) Toss a coin three times; ={H,T}HHH,HHT,HTH, HTT,THH,THT,TTH,TTT}
Toss a coin three times and observe the number of H's; ={H,T}0,1,2,3} Toss a coin until we obtain two H's in succession or two tails (countably not necessarily in succession); ={H,T}HH,TT,THH,THT,HTT,HTHH,HTHT} Queue size at a railway reservation counter; ={H,T}0,1,2,…}; (countably } Time taken (countably in minutes) to download a website; ={H,T}t:0<t<}
Examples Toss a coin and A is the event that outcome is "H“; A={H,T}H} Throw a dice. Let A be the event that outcome is an even number; A={H,T}2,4,6} Life of a bulb is more than 1000 hrs; A={H,T}t:1000<t<} Toss a coin until we obtain a "H". A be the event that number of tosses is more than 3; A={H,T}TTTH,TTTTH,…}; (countably } No customer in the queue at the railway reservation counter; A={H,T}0}
When is an event said to occur? An event A is said to occur if outcome of the random experiment under consideration has a description that is a member of A.
Complement of A, denoted by A c , is an event which occurs whenever A does not occur Let A 1
2 ,…}; (countably ,A k be k events. Then be an event which occurs when at least one of the events A i occurs. k i i
1
Further, be an event which occurs when all of the events A i occurs. A-B=AB c be an event which occurs when A occurs and B does not occur. Obviously A c =-A. k i i
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In general, the sample space for n repetitions of the random experiment is …}; (countably (countably n times). The occurrence of event A implies the occurrence of event B, i.e., whenever event A occurs event B also occurs. We denote it by AB. (countably In set theory AB)
Two events A and B are said to be mutually exclusive if AB=. (countably In set theory A and B are called disjoint if AB=) Ex: In throwing a dice, if A is the event that outcome is an even number and B is the event that outcome is an odd number, then A and B are mutually exclusive events.
Axioms of a Probability Measure 0 P(countably A)1 for every AB P(countably )= For two mutually exclusive events A and B (countably A,BB, AB=) P(countably AB)=P(countably A)+P(countably B) For pair-wise mutually exclusive events A 1 ,A 2 , …}; (countably (countably A i B i and A i A j = ij)
(countably additive) A function P satisfying above axioms is called a Probability Measure. P(countably A): A measure of how confident we are that the outcome will be in A. Ex: Prove that P(countably )=0. The triplet (countably , B, P) is called the probability space i i i i P ( A ) P ( A )
