351K EE Probability, Statistics, and Random Processes Instructor:
Shakkottai/Vishwanath Homework 3  Solution SPRING 2008
{shakkott,sriram}@ece.utexas.edu
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 0.4. If a
mosquito lands, it will bite you with probability 0.4, and it will never bother you with
probability 0.6, independently of other mosquitoes. What is the expected time between
successive bites? Solution : At any second the probability that a mosquito bit you is 0.4
0.4 = 0.16. So, the time T in seconds between successive bites is a geometric random
variable with parameter p = 0.16. It follows that E[T ] = 1/p = 6.25 seconds.
Problem 2 Let X1 , ..., Xn 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: 2 E[(X1 +
....
+ Xn )2 ] = cE[X1 ] + dE[X1 ]2 . Solution : 2 Using the formula
var(X) = E[X 2 ]  E[X] , we have E (X1 + + Xn )2 = var(X1 + + Xn ) + E[X1 + + Xn ] = n
var(X1 ) + (nE[X1 ])2 2 = nE[X1 ]  n E[X1 ] 2 2 + n2 E[X1 ] 2 2 2 = nE[X1 ] + n(n  1)
E[X1 ] . Thus,c = n and d = 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 pX,Y . Solution : The probability of any set
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 Spring '07
 BARD
 Probability, Probability theory, Pens, Oscar, supply room

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