6.262.PS2

6.262.PS2 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 – Discrete Stochastic Processes Problem Set #2 Issued: February 12, 2010 Due: February 19, 2010 1. For each of the sequences of random variables below, determine if it converges to zero in distribution as and also determine if it converges to zero in mean-square as . n →∞ n a) The sequence of random variables with the property that n X n ( 1 )1 /, P ( X 0 )11 / . n PX n n = == = b) The sequence of random variables with the property that n Y n () 1 / , P ( Y 0 ) 1 1 / n PY n n n . = = c) The player’s wealth in the “Double or Quarter” game. (See the last problem on Pset #1.) n W 2. We say that a discrete-time random process , 0,1,2, n Xn = " has stationary increments if ( has the same distribution as ) nm n XX + 0 m for all We say that a discrete-time random process , 0 . 0,1,2, n = " has pair-wise independent increments if and ( nl n + ) mr m + are independent for all A process has independent increments if it has pair-wise increments and three-wise independent increments and four-wise independent increments, etc. (Something worth thinking about: why are pair-wise independent increments not enough?) Note that these definitions are equivalent to those in Lecture 3 but should be a bit easier to work with. , , , 0 with . nmlr n m r ≥≥ + a) A random walk is a random process given by 0, n Sn 0 , n nk k SY = = 1
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where is a set of iid random variables. (The Bernoulli process is an example of a random walk.) Does every random walk have stationary increments? Does every random walk have independent increments? , 0 k Yk We say that a continuous-time random process has stationary increments if () t 0 , Xt (( ) ( ) ) τ +− has the same distribution as ) ( 0 ) ) XX for all t, 0. We say that a continuous-time random process has pair-wise independent increments if t 0 , ) ( ) ) and ) ) Xs X υ are independent for all , , , 0 with t . ts s υυ ≥≥ + Then, analogously to the discrete case, a process has independent increments , if it has k - wise independent increments for any integer k . (These definitions are also equivalent to those in Lecture 3 but, again, they should be a bit easier to work with.) b) Consider the continuous-time renewal process X(t) = (, ) , 0 , Ber p t t δ > obtained from
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.PS2 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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