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Unformatted text preview: EE 131A Comprehensive project Probability Friday, November 5, 2010 Instructor: Lara Dolecek Due: Friday, December 3, 2010 Reading: Chapters 2 through 8 of Probability, Statistics, and Random Processes by A. Leon-Garcia 100 points total In this project we will further analyze random variables and their various properties. We will also investigate how are random variables used to model practical systems. Each part will have a combination of MATLAB programming, mathematical analysis and technical writing. You will be graded on all three components. When producing your plots clearly indicate the x-axis, the y-axis and what is being plotted (using legends, title etc.). You may need to rescale x-axis to ensure that your plot is showing the right quantity. Make sure to attach in the appendix of your project report all MATLAB programs that you used to generate the data. 1. (10 pts) Tossing a fair die. Suppose you have a 6-sided fair die. (a) Write a MATLAB program to simulate the tossing of a fair die, for t =10, 50, 100, and 1000 tosses. Based on the simulation, what is the probability of obtaining an even number? (b) Suppose X is a random variable denoting the outcome of a die toss. What type of distribution does X have ? Based on the analysis, what is the probability that X has even value ? (c) Refer back to part (a). Does it agree with the theoretical result in (b) ? You may find useful the MATLAB function rand that generates a uniform random value in the (0 , 1) interval. 2. (10 pts) Generation of binomial RVs. (a) Write a MATLAB program to simulate a X binomial( n,p ) random variable for ( n,p ) = (10 , . 5), (10 , . 25), (10 , . 1) and (100 , . 01). Use t = 1000 trials in each case to generate X , and plot pmf in each case. (b) What value of k maximizes P ( X = k ) for a general binomial( n,p ) ? (c) What value of k maximizes P ( X = k ) in your simulations ? Contrast the result in (b) with your experimental findings from part (a). 1 3. (10 pts) Generation of Poisson RVs. (a) Write a MATLAB program to generate realizations of a Y Poisson (1) random variable by approximating it with Z binomial(100 , . 01) random variable with t = 1000 trials. Plot pmf of Z . What is the average value of Z based on your results? (b) Write a MATLAB program to generate Poisson (1) using t = 1000 samples. Plot the resulting pmf. (c) Compare the theoretical value of P [ Y = 3] for the Poisson random variable with = 1 to the estimated value obtained from the simulations in the previous parts....
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