hw3_sol - EE 351K Probability Statistics and Random...

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EE 351K Probability, Statistics, and Random Processes SPRING 2009 Instructor: Shakkottai/Vishwanath { shakkott,sriram } @ece.utexas.edu Homework 3 - Solution Problem 1 You are visiting the rainforest, but unfortunately your insect repellent has run out. As a result, at each second, a mosquito lands on your neck with probability 0.5. If a mosquito lands, it will bite you with probability 0.3, and it will never bother you with probability 0.7, independently of other mosquitoes. What is the expected time between successive bites? Solution : At any second the probability that a mosquito bit you is 0 . 5 · 0 . 3 = 0 . 15 . So, the time T in seconds between successive bites is a geometric random variable with parameter p = 0 . 15 . It follows that E [ T ] = 1 /p = 20 / 3 seconds. Problem 2 Let X 1 , ..., X n be independent, identically distributed random variables with common mean and variance. Find the values of c and d that will make the following formula true: E [( X 1 + .... + X n ) 2 ] = c ( E [ X 1 ]) 2 + dE [( X 1 ) 2 ] . Solution : E ± ( X 1 + ··· + X n ) 2 ² = E ± n X i =1 X 2 i + X i 6 = j X i X j ² = n X i =1 E ± X 2 i ² + X i 6 = j E ± X i X j ² (by linearity of expectation) = n X i =1 E ± X 2 i ² + X i 6 = j E ± X i ² E ± X j ² (by independence) = n X i =1 E ± X 2 1 ² + X i 6 = j E ± X 1 ² E ± X 1 ² (by identically distributed) = nE ± X 2 1 ² + n ( n - 1)( E ± X 1 ² ) 2 Where the last equality follows from there are n terms in the first summation and n ( n - 1) terms in the second summation. Thus, d = n and c = n ( n - 1) . Problem 3 A class of n students takes a test in which each student gets an A with probability p , a B with probability q , and a grade below B with probability 1 - p - q , independently of any other student. If X and Y are the numbers of students that get an A and a B, respectively, calculate the joint PMF p X,Y . Solution :
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hw3_sol - EE 351K Probability Statistics and Random...

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