ST115 2017-2018 Lecture notes - Copy (7).pdf

ST115 2017-2018 Lecture notes - Copy (7).pdf - ST115...

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ST115 Introduction to Probability 2017/2018 Dr Zorana Lazic March 5, 2018 1
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2 Contents 1 Probability 4 1.1 Random Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Axioms of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Properties of Probability Measures . . . . . . . . . . . . . . . . . . . . . . . 7 2 How to Count 14 2.1 Fundamental counting principle . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Conditional Probability 18 3.1 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Law of Total Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Independence 23 4.1 Independence of events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Random Variables 25 5.1 Definition of Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.2 Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3.1 Cumulative Distribution Function of a Discrete Random Variable . . 29 5.4 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.5 Common Discrete Probability Distributions . . . . . . . . . . . . . . . . . . 32 5.5.1 Bernoulli distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.5.2 Indicator random variable . . . . . . . . . . . . . . . . . . . . . . . . 32 5.5.3 Binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.5.4 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.5.5 Geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.6 Common Continuous Probability Distributions . . . . . . . . . . . . . . . . 34 5.6.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.6.2 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.6.3 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6 Expectation 36 6.1 Expectation of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 Variance of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 Moment generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.4 Common Discrete Probability Distributions (revisited) . . . . . . . . . . . . 43 6.4.1 Bernoulli distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4.2 Binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.4.3 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.4.4 Geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.5 Common Continuous Probability Distributions (revisited) . . . . . . . . . . 45 6.5.1 Uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.5.2 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.5.3 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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3 7 Transformation of a Random Variable 46 7.1 Transformation of a Discrete Random Variable . . . . . . . . . . . . . . . . 46 7.2 Transformation of a Continuous Random Variable . . . . . . . . . . . . . . 46 8 Joint Distributions 48 8.1 Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.2 Discrete Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8.3 Continuous Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 50 8.4 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.5 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.6 Distribution of a sum of independent random variables . . . . . . . . . . . . 57 8.7 Covariance and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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4 1 Probability 1.1 Random Experiments Definition 1.1.1. Random experiment A random experiment is a process (or a trial) for which the outcome cannot be predicted with certainty. In order to study random experiments we need to construct a mathematical model. First we need to identify the possible outcomes of a random experiment. Definition 1.1.2. Sample space The sample space Ω of a random experiment is the set of all possible outcomes of that experiment. The elements of the sample space are called sample points. Definition 1.1.3. Event An event is a set of outcomes of a random experiment. Each event is a subset of the sample space of the experiment. Events are usually denoted by capital Roman letters e.i. A, B, C, . . . . If the outcome of an experiment belongs to a certain event, we say that that event has occurred. The sample space Ω is an event which always occurs, and the empty set is an event which never occurs. We look at some examples of random experiments, sample spaces and events: Example 1.1.1. 1. Tossing a coin. Ω = { H, T } (H = head, T = tail).
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  • Spring '17
  • Jane Kovac
  • Probability theory

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