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 one-cell 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 Inter-arrival times of customers to the system 13 An Example using both Discrete and Continuous RVs Background: A one-cell 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; inter-arrival 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) Non-increasing 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 Hyper-Exponential 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 n-th moment: For discrete RVs: For continuous RVs: The n-th 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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