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

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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. , if 4. 5. 6. 7. 8. , Definition of PMF: (Using the abbreviated event notation) where are the points at which of a discrete random variable Properties of PMF: 1. 2. 3. ∑ 4. If , , if ∑ is a continuous random variable, then . jumps. 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? Solution. Let Assume , , define random variable , then the PMF on as is Hence 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? Solution. Let , . Define random variable as Assume , the PMF on is () where we have written () as . The main reasoning involved in getting the above result is to observe: (i) there are totally ( ) outcomes that has heads and tails; (ii) for each such outcome the probability for observing it is . Hence ∑ () ∑ ∑( ∑ ) ∑( ) † and 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? Solution. Let as Assume † , then , then the PMF An easier way is to treat the . is a countably infinite set. on . Define random variable is defined here as the sum of of the previous question and exploit the linearity of : where we have written as . Hence ∑ ∑ ∑ ∑ ( ) and 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? Solution. Let , . Define random variable as Assume , then the PMF on is ( where we have written ) as . We now proceed to compute the expected value. First we extract an identity from the fact that ∑( ) ∑ ∑ ∑ ∑ hence ∑ is a PMF: A simpler way of deriving the expected value is to start with viewing a negative binomial r.v. as the sum of geometric r.v.’s. 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 . Solution. (a) The following manipulation is purely technical: as , ( ( Now it remains to prove that → )( ) ( ) ) . This is because ( )( ) ( )→ and q.e.d. (b) This part gives you the practical explanation of the term “Poisson approximation to Binomial”. It also shows how possibly tedious can a statistical problem be and by this way motivates you to use software packages such as the free R language (cran.r-project.org, type ?dbinom and ?dpois with the question mark to see help files on how to use these built-in pdf functions, if you decide to give it a try). ( ) ( ) ( ) ( ) 6. (a) Show that ∫ ( ) and find the integration. (b) Show that ∫ and find the integration. (c) Show that , , and hence for : integer. (d) Find the normalizing constant such that the function is a valid PDF. Solution. (a) () ∫ ∫ ( ∫ Let ∫ () ⁄ ∫ ) () , then ⁄ ∫ ∫∫ ∫ ∫ ∫ √ (b) ∫ (c) ∫ ∫ [ ∫ ∫ Since and for each integer , we will have if we have th . Then, by principle of induction (the 5 Peano’s axiom for natural numbers), we can have the induction hypothesis for real. (d) Since the integration of ∫ over the entire real axis must be 1, we shall have ∫ √ √ ∫ √ √ √ 7. (a) Write down the PDF of , derive the expectation and variance. ( (b) Write down the PDF of ), its expectation and variance. (c) Write down the CDF of , its expectation and variance. (d) Derive the MGF of . (Hint: Solution. (a) The pdf is , . The trick of remembering this is to observe the resemblance of the kernel to the function and to understand how the additional scaling parameter is incorporated. ∫ ∫ ∫ ∫ [ (b) The pdf is the just the ( ) pdf: () (c) The pdf is just the pdf: Thus ∫ ∫ (d) ∫ √ √ ∫ √ ∫ ...
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.

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