E_tnlask - J. Virtamo 38.3143 Queueing Theory / Probability...

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Unformatted text preview: J. Virtamo 38.3143 Queueing Theory / Probability Theory 1 ESSENTIALS OF PROBABILITY THEORY Basic notions Sample space S S is the set of all possible outcomes e of an experiment. Example 1. In tossing of a die we have S = { 1 , 2 , 3 , 4 , 5 , 6 } . Example 2. The life-time of a bulb S = { x ∈ R | x > } . Event An event is a subset of the sample space S . An event is usually denoted by a capital letter A, B, . . . If the outcome of an experiment is a member of event A , we say that A has occurred. Example 1. The outcome of tossing a die is an even number: A = { 2 , 4 , 6 } ⊂ S . Example 2. The life-time of a bulb is at least 3000 h: A = { x ∈ R | x > 3000 } ⊂ S . Certain event : The whole sample space S . Impossible event : Empty subset φ of S . J. Virtamo 38.3143 Queueing Theory / Probability Theory 2 Combining events Union “ A or B ”. A ∪ B = { e ∈ S | e ∈ A or e ∈ B } A B A B È Intersection (joint event) “ A and B ”. A ∩ B = { e ∈ S | e ∈ A and e ∈ B } A B Ç A B Events A and B are mutually exclusive , if A ∩ B = φ . Complement “not A ”. ¯ A = { e ∈ S | e / ∈ A } A A Partition of the sample space A set of events A 1 , A 2 , . . . is a partition of the sample space S if 1. The events are mutually exclusive, A i ∩ A j = φ , when i negationslash = j . 2. Together they cover the whole sample space, ∪ i A i = S . A 1 A 2 J. Virtamo 38.3143 Queueing Theory / Probability Theory 3 Probability With each event A is associated the probability P { A } . Empirically, the probability P { A } means the limiting value of the relative frequency N ( A ) /N with which A occurs in a repeated experiment P { A } = lim N →∞ N ( A ) /N N = number of experiments N ( A ) = number of occurrences of A Properties of probability 1. 0 ≤ P { A } ≤ 1 2. P {S} = 1 P { φ } = 0 3. P { A ∪ B } = P { A } + P { B } - P { A ∩ B } A B A B È 4. If A ∩ B = 0, then P { A ∪ B } = P { A } + P { B } If A i ∩ A j = 0 for i negationslash = j , then P {∪ i A i } = P { A 1 ∪ . . . ∪ A n } = P { A 1 } + . . . P { A n } 5. P { ¯ A } = 1- P { A } 6. If A ⊆ B , then P { A } ≤ P { B } J. Virtamo 38.3143 Queueing Theory / Probability Theory 4 Conditional probability The probability of event A given that B has occurred. P { A | B } = P { A ∩ B } P { B } ⇒ P { A ∩ B } = P { A | B } P { B } A B P{A B} Ç P{B} J. Virtamo 38.3143 Queueing Theory / Probability Theory 5 Law of total probability Let { B 1 , . . ., B n } be a complete set of mutually exclusive events, i.e. a partition of the sample space S , 1 . ∪ i B i = S certain event P {∪ i B i } = 1 2 . B i ∩ B j = φ for i negationslash = j P { B i ∩ B j } = 0 Then A = A ∩ S = A ∩ ( ∪ i B i ) = ∪ i ( A ∩ B i ) and P { A } = n summationdisplay i =1 P { A ∩ B i } = n summationdisplay i =1 P { A | B i } P { B i } Calculation of the probability of event A by conditioning on the events...
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E_tnlask - J. Virtamo 38.3143 Queueing Theory / Probability...

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