Example Class 4 solution - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I, FALL 2010 EXAMPLE CLASS 4 Distribution Elements of Theory Abbreviated Event Notation Usually we abbreviate the description of the event/set to simply as so that we can also abbreviate the probability notation as or even simpler as . This is primarily because the two probability measures and are consistent due to the property of the random variable . Definition of CDF: , Properties of CDF: 1. 2. , if 3. 4. 5. 6. 7. 8. , Definition of PMF: (Using the abbreviated event notation) where are the points at which of a discrete random variable jumps. Properties of PMF: 1. , 2. , if 3. 4. If is a continuous random variable, then .
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Definition of PDF: Properties of PDF: 1. 2. 3. 4. Gamma Function and Beta Function: , . . Exercises 1. The Random Experiment is “Tossing a coin (not necessarily a fair one) once. And if the coin turns out a head, then you earn 1 dollar, otherwise, 0 dollar.” You are interested in how many dollars you will win after tossing it once. Define an appropriate random variable to describe the experiment whose state space should be {0,1} and derive the PMF on its state space. (Hint: Make assumptions in term of parameters whenever needed, e.g., the a priori probability governing the chance that the coin will be turning out a head.) What’s the expected value of dollars you will win?
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