Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables


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1 TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics Yuming Jiang 1 Some figures taken from the web. Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random Variables Distribution Functions Mean, Variance, Moments 2
2 Basic Concepts Experiments, Outcomes, Sample Space, Events Experiment A procedure that can be repeated and has a well defined set of outcomes Ex: Tossing a coin. Outcome (denoted by ω) The result of an experiment. Ex: In one experiment of tossing a coin, the outcome is H. Sample Space (denoted by Ω) The set of all possible outcomes of an experiment. Ex: The possible outcomes of tossing a coin are H and T. So, the sample space of tossing a coin is {H, T}. Event An event is one or more outcomes (or a set of outcomes) of an experiment. Mathematically, it is a subset of the sample space. Ex: {H}, or {T}, or {H, T} 3 Basic Concepts Combination of Events Intersection of Events Ex: Two events {T} and {H, T}. Their intersection is {T}. Union of Events Ex: Two events {T} and {H, T}. Their union is {H, T}. Mutually Exclusive Events are mutually exclusive, if their intersection is the empty set. Exhaustive Events are mutually exclusive, if their union is the sample space. 4
3 Basic Concepts Combination of Events 5 Basic Concepts Probability of an Event Consider an event A. The probability of A, P(A), measures the relative frequency in which event A happens out of all possible outcomes. Ex: Let A={T} in the tossing coin experiment. P(A)=0.5. Properties 6
4 An Example Experiments: Finding the number of students attending a TTM4155 class. The Outcome (of the experiment on ): 16 students attended the class. The Sample Space The total registered students to the course is 16. {0, 1, 2,, 16} Event A The number of students in the class (on any course day) is more than 10. {11, 12,, 16} At the end of the semester, we can calculate P{A} as P{A} = number of days A is satisfied / total number of course days 7 Another Example Background: A onecell mobile phone network exclusively designed for use by customers in the class room of TTM4155 during the class time. Experiments: Finding the number of potential customers of the network. The Sample Space The total registered students to the course is 16. On any course day, there is 1 lecturer. {0, 1, 2,, 16, 17} The Outcome (of the experiment on ): 17, i.e. 1 lecturer + 16 students attending the class. 8 Event B The number of customers using the network at the same time is more than 10. {11, 12,, 16, 17} How to calculate P{B}? What more information is needed? If the network is positioned to support only 10 phone connects at one time, what is the blocking probability of a call made by these customers?
5 Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random Variables Distribution Functions Mean, Variance, Moments 9 Random Variables Related to events Each outcome in the sample space, i.e. ω in Ω, is associated with a real number X(ω) which is called a random variable. It is a mapping from the sample space to the set of all real numbers, i.e. {X=x}, where x is a real number, is an event. The probability of this event is denoted by P(X=x). 10
6 Example 11 Discrete Random Variables A random variable X is discrete if there is a discrete set S X in R such that P{X є S X } =1. Here, a set S X is called discrete if it is finite, S X = {x 1, x 2,, x n }, or countably infinite, S X = {x 1, x 2,, }. The set S X is also called the value set. Examples: Number of tails in tossing coin experiments Number of students in class Number of customers in system 12
7 Continuous Random Variables Random variable X is continuous if there is an integrable function f X in R such that for all x є R Here, the function f X is called the probability density function (pdf) The set S X is also called the value set. Examples: Temperature of a day Arrival times of students to the class Interarrival times of customers to the system 13 An Example using both Discrete and Continuous RVs Background: A onecell mobile phone network exclusively designed for use by customers in the class room of TTM4155 during the class time. Description of Customers of the Network Customer arrivals: arrival time of each student / lecturer to the class; interarrival times of students/lecturer to the class (continuous RVs) Numbers of customers in the network: number of students/lecturer in the room; number of students/lecturer in the room seen by an arriving student, etc. (discrete RVs). 14
8 Contents Basic Concepts Random Variables Distribution Functions Discrete Random Variables Continuous Random Variables Mean, Variance, Moments 15 Distribution Functions Cumulative distribution function (cdf) F X (x) F X (x) = P(X x) Properties of F X (x) 16 Well defined for both discrete and continuous RVs.
9 Distribution Functions (cont ) Complimentary cumulative distribution function (ccdf) F c X (x) F c X (x) = P(X>x) Properties of F c X (x) F c X (x) =1 F X (x) Nonincreasing F c X ( ) = 1 F c X (+ ) = 0 17 Well defined for both discrete and continuous RVs. Distribution Functions (cont ) Probability mass function (pmf) P X (x) P X (x) = P(X=x) Well defined for discrete RVs; If X is a continuous RV, P X (x) =0. Probability density function (pdf) f X Well defined for continuous RVs 18
10 Example 19 Examples Discrete random variables Bernoulli Geometric Binomial Poisson Pascal 20 Continuous random variables Uniform Exponential HyperExponential Erlang Gaussian Pareto
11 Contents Basic Concepts Random Variables Distribution Functions Mean, Variance, Moments 21 Mean (/Expectation) Also called expectation, denoted as E[X]. Defined as: For discrete RVs: For continuous RVs: 22
12 Variance Defined as It can be proved: 23 Moments The nth moment: For discrete RVs: For continuous RVs: The nth central moment: For discrete RVs: For continuous RVs: 24
13 Example Bernoulli, Binomial Bernoulli Discrete random variable Take value 1 with success probability p and value 0 with failure probability q = 1 p The probability (mass) function: Binomial: pmf and cdf Mean: p Variance: pq Binomial (n, p) The sum of n independent, identically distributed (i.i.d.) Bernoulli random variables results in the Bernoulli distributed random variable. Mean: np Variance: npq 25 Example Exponential Exponential: pdf and cdf Probability density function λ λx Mean: 1/λ Variance: 1/ λ 2 26
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