lecture5.1

# lecture5.1 - Random Variables Given a random experiment and...

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Unformatted text preview: Random Variables Given a random experiment and the sample space Ω , we have already discussed how to assign a probability to subsets of the sample space. Yet in many occasions, we are interested in a random quantity that is a function of the outcome, instead of the outcome itself. Summarizing the outcome by a real number allows easier mathematical treatments. () January 23, 2011 1 / 31 Random Variables Given a random experiment and the sample space Ω , we have already discussed how to assign a probability to subsets of the sample space. Yet in many occasions, we are interested in a random quantity that is a function of the outcome, instead of the outcome itself. Summarizing the outcome by a real number allows easier mathematical treatments. We will use Ω for sample space to avoid some confusion. () January 23, 2011 1 / 31 Random Variables: Example Let the random experiment be “flipping 3 fair coins”. Let Y = “number of heads observed”. Then, the entity “ Y ” translate each outcome(sample point) of the experiment into a number. We regard “Y=1”, for instance, as an event containing all sample points that are translated by Y into value 1. Apparently, in this examples, “ Y = 1” is the same as the event F = { HTT , THT , TTH } which is a subset of the sample space Ω = { ( TTT , HTT , THT , TTH , HHT , HTH , THH , HHH } . () January 23, 2011 2 / 31 Random Variables: Example Because three coins are fair, it is justified to assign equal probability to every single outcome (as event). Hence, P ( F ) = favourable All Possible = 3 8 . Because the event F is equivalently represented by “ Y = 1”, we also write P ( Y = 1 ) = 3 8 . () January 23, 2011 3 / 31 Random Variables: Example In the same vein, we have P ( Y = ) = 1 / 8; P ( Y = 1 ) = 3 / 8; P ( Y = 2 ) = 3 / 8; P ( Y = 3 ) = 1 / 8. The event Y = 4, for instance, in this experiment is empty. We do not bother to tell you P ( Y = 4 ) = 0 in general. From the fact that 1 / 8 + 3 / 8 + 3 / 8 + 1 / 8 = 1, we can tell that we have exhausted all possibilities. () January 23, 2011 4 / 31 Random Variables: Definition A random variable is a real valued function defined on a sample space: Y : Ω → R . We usually use CAPITAL letters near the end of alphabet such as X , X 1 , Y and so on to denote random variables. I will follow this rule as much as possible, but do not take anything as granted. Do read the content before determining the nature of a notation. () January 23, 2011 5 / 31 Random Variables: Definition A random variable is a real valued function defined on a sample space: Y : Ω → R . We usually use CAPITAL letters near the end of alphabet such as X , X 1 , Y and so on to denote random variables....
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lecture5.1 - Random Variables Given a random experiment and...

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