Probability and Stochastic Processes

Probability and Stochastic Processes - Probability 1 Random...

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Probability 1 Random experiments A random experiment is an experiment whose outcome is not known in advance, but for which the set of all possible individual outcomes is known. An experiment consists of a procedure and observations. There is some uncertainty in what will be observed; otherwise, performing the experiment would be unnecessary. Some examples of experiments include: 1. Flip a coin. Does it land on heads or tails? 2. Walk to a bus stop. How long do you wait for the arrival of a bus? 3. Give a lecture. How many students are seated in the fourth row? For the most part, we will analyze models of actual physical experiments. We create models because real experiments generally are too complicated to analyze. Consequently, it is necessary to study a model of the experiment that captures the important parts of the actual physical experiment. Example An experiment consists of the procedure, observation and model: Procedure: Flip a coin and let it land on a table. Observation: Observe which side (bead or tail) faces you after the coin lands. Model: Heads and tails are equally likely, The result of each flip is unrelated to the results of previous flips. W Two experiments with the same procedure but with different observations are different experiments. Example Flip a coin three times, observe the sequence of heads and tails. Flip a coin three times, observe the number of heads. These two experiments have the same procedure: flip a coin three times. They are different experiments because they require different observations. W We will describe models of experiments in terms of a set of possible experimental outcomes. In the context of probability, we give precise meaning to the word outcome. An outcome of an experiment is any possible observation of that experiment. As a result, we define the universal set of all possible outcomes in probability terms, we call this universal set the sample space. An event is a set of outcomes of an experiment. 1.1 A form of model building in the language of probability Real World Mathematical Terminology 1. All possible outcomes of a random experiment Sample space 2. A single outcome (simple event) A sample point ϖ 3. A set of outcomes (event) A subset of the sample space A 1
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Example Suppose we roll a six sided die and observe the number of dots on the side facing upwards. We can label these outcomes 1,2, ,5,6 i = L where i denotes the outcome that i dots appear on the up face. The sample space is { } 1,2, ,5,6 S = L . Each subset of S is an event. Examples of events are: The event { } 1 roll 4 or higher 4,5,6 E = . The event { } 2 The roll is even 2,4,6 E = . { } 3 The roll is a square of an integer 1,4 E = . W 1.2 Operations with events Real World Mathematical Terminology Event A occurs A ϖ A does not occur A ϖ or C A ϖ At least one of A or B occurs A B ϖ Both A and B occur A B ϖ A occurs and B docs not occur C A B ϖ or \ A B ϖ A implies B B A The sure event The impossible event φ A and B are mutually exclusive A B φ =
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