5 Example Class 4

5 Example Class 4 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: 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. 3. 4. 5. 6. 7. 8. , , if , 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. 3. ∑ 4. , , if ∑ If is a continuous random variable, then . 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? 2. The Random Experiment changes to “Tossing a coin (not necessarily a fair one) times; if the coin turns out a head, then you earn 1 dollar, otherwise, 0 dollar.” You are still interested in how many dollars you will win after finishing the experiment, i.e., tossing times. Define a new random variable to describe the experiment whose state space should be and derive the PMF on its state space. Assume . What’s the expected value of dollars you will win? What’s the probability that you will win at least some money? 3. The Random Experiment changes again, to “Tossing a coin (not necessarily a fair one) indefinitely until it turns out a head. The number of dollars you will win is equal to the number of tosses it takes to see the first head turning out.” You are still interested in how many dollars you will win after finishing the experiment, which equals the number of tosses until you see the first time a head turns out. Define a new random variable to describe the experiment whose state space should be and derive the PMF on its state space. Assume . What’s the expected value of dollars you will win? What’s the probability that you will win at least dollars? 4. The Random Experiment changes yet again, to “Tossing a coin (not necessarily a fair one) indefinitely until heads have been observed. The number of dollars you will win is equal to the total number of tosses until you see the th head turning out.” You are still interested in how many dollars you will win after finishing the experiment, which equals the number of tosses until you see the th head. Define a new random variable to describe the experiment whose state space should be and derive the PMF on its state space. Make necessary parametric assumptions when needed. Assume . What’s the expected value of dollars you will win? 5. (a) Verify that Poisson distribution can approximate Binomial distribution when the number of Bernoulli trials is very large and is very small, while the mean remain finite. To be precise, suppose has a binomial distribution with parameters and . If and as then (b) , , , and . Compare the values of and . 6. (a) Show that ∫ (b) Show that ∫ (c) Show that ( ) and find the integration. and find the integration. , , and hence for : integer. (d) Find the normalizing constant such that the function is a valid PDF. 7. (a) Write down the PDF of (b) Write down the PDF of (c) Write down the CDF of (d) Derive the MGF of . (Hint: ( , derive the expectation and variance. ), its expectation and variance. , its expectation and variance. ...
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.

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