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(solution) Stuck at Question 5, I realize that a need a E(X/Y) and that pdf


Stuck at Question 5, I realize that a need a E(X/Y) and that pdf is linear, but don't know how to proceed.

Thank You!



Foundations of Stochastic Processes (40256) ? G. Caire ? December 10, 2015 1 Module: Foundations of Stochastic Processes (40256) ? WS 2015-2016 HOMEWORK SET #3

 

This homework is due on Thursday 17.12.2015, ***at the beginning of the 12:00

 

Lecture***. Late delivery is not accepted (please don?t miss the lecture to try to ?nish

 

the homework and deliver it at the end of the Lecture, since this will not be accepted). 1. A method to generate a random variable X with assigned cdf FX (x) is as follows: let U ?

 

?1

 

Uniform[0, 1] denote a uniformly distributed random variable on [0, 1]. Then, X = FX (U )

 

has the desired cdf.

 

?1

 

a) Prove that the above statement is true, that is, show that P(FX (U ) ? x) = FX (x). b) Use the above method in order to generate (using Matlab or your favorite numerical

 

computation tool) a random number generator X ?exponential with parameter ?. For ? = 1,

 

plot the empirical cdf using N = 1000 samples. [INCLUDE YOUR PLOT IN THE

 

HOMEWORK PAPER].

 

Hint: in order to plot an empirical cdf of a set of values {x1 , . . . , xN } you need to plot the

 

staircase function

 

N

 

1

 

(N )

 

F (x) =

 

I{x?xi }

 

N

 

i=1 Notice that for any given x the number of indicator functions equal to 1 in the sum above

 

counts the number of values xi in the set that are less or equal to x. This is precisely the

 

de?nition of a cumulative distribution function. Now, in order to plot this function e?ciently,

 

it is su?cient to sort the set of values in non-decreasing order with the command ?sort? in

 

Matlab. Letting x[1] , x[2] , . . . , x[N ] denote the sorted sequence, and letting {yi = (i ? 1)/N :

 

i = 1, . . . , N } and {zi = i/N : i = 1, . . . , N }, it is su?cient to create a vector x of length 2N

 

with components

 

x[1] , x[1] , x[2] , x[2] , x[3] , x[3] , . . . , x[N ] , x[N ]

 

and a vector y of length 2N with components

 

y1 , z1 , y2 , z2 , y3 , z3 , . . . , yN , zN

 

and use the command ?plot? with x on the x-axis and y on the y-axis. In this way, you will

 

create the staircase function corresponding to the empirical cdf F (N ) (x) given above.

 

2. Consider a communication link where packets of information bits are sent through a data

 

link that operates at the rate of R bit/s. At the output, a packet is either received correctly

 

or it is completely lost (erased), with probability p. The erasure events are independent and

 

identically distributed.

 

Consider the following simple repetition protocol. A packet is repeated until it is correctly

 

received. When a packet is correctly received, the transmitter removes it from its transmission

 

queue and moves on to transmit the next packet in the queue bu?er. It is assumed that an

 

in?nite sequence of packets are in the bu?er, such that the queue is never empty. Foundations of Stochastic Processes (40256) ? G. Caire ? December 10, 2015 2 a) Compute the expected number of transmissions in order to correctly receive a packet.

 

b) Compute the system rate, expressed as the average number of correctly received bits per

 

second.

 

3. A communication channel works as follows: in order to transmit a binary symbol B ? {0, 1},

 

we repeat this symbol 5 times, i.e., in order to transmit B = 0 we transmit 00000 and in

 

order to transmit B = 1 we transmit 11111. The channel introduces random errors by ?ipping

 

independently the symbols with probability p (this means that any transmitted bit 0 or 1 is

 

turned into its complement 1 or 0, respectively, with probability p, and it is left unchanged

 

with probability 1?p). Then, after this independent bit-?ipping, because of some malfunction

 

of the receiver, the bits are erased independently with probability . For example, suppose

 

that we transmit B = 0, i.e., we send through the channel the sequence 00000. If the channel

 

?ips the 1st and the 4th bit and the receiver erases the 3rd bit, then the received sequence at

 

the channel output will be 10x10, where x denotes an erasure.

 

?

 

The receiver determines the value of an estimate B of B as a function of the channel output

 

sequence. In particular, if it receives more zeros than ones in the non-erased symbols, it

 

?

 

?

 

decides for B = 0. Otherwise, it decides for B = 1. In the case of a tie (or if all symbols

 

are erased), it ?ips a fair coin and decides with equal probability for 0 or 1. Assuming

 

that that B is uniformly distributed on {0, 1}, ?nd an expression for the probability of error

 

?

 

Pe = P(B = B).

 

4. The decay time of a radioactive isotope is exponentially distributed. You can measure the

 

decay time Y , and you have two hypotheses, corresponding to two di?erent isotopes:

 

H0 : Y ? ?0 e??0 y Iy?0

 

H1 : Y ? ?1 e??1 y Iy?0

 

The in nature, the probability of ?nding isotope H0 is 10%, and probability of ?nding isotope

 

H1 is 90%.

 

a) Find the optimal Bayesian decision rule that minimizes the average probability of error in

 

deciding which isotope is found while observing Y .

 

b) Find the expression of the corresponding minimum error probability.

 

5. Consider the observation model

 

Y =X +Z

 

where Z is Laplacian, i.e., it has pdf fZ (z) = ? e??|z| and X is binary, taking values +1 and

 

2

 

?1 with equal probability. X and Z are statistically independent.

 

a) Find the linear MMSE estimator Xlin of X given Y , and the corresponding Mean-Square

 

Error.

 

b) Find the optimal MMSE estimator Xopt of X given Y , and the corresponding Mean-Square

 

Error.

 

6. Bonus question: THIS GIVES YOU EXTRA POINTS!! Let X and Y be i.i.d.

 

? N (0, 1). Find

 

min{|X|, |Y |}

 

E

 

max{|X|, |Y |} Foundations of Stochastic Processes (40256) ? G. Caire ? December 10, 2015 3 Hint: Notice that the Gaussian random vector (X, Y )T is rotationally invariant, that is, for

 

any orthogonal (rotation) 2×2 matrix R we have that R(X, Y )T has the same pdf of (X, Y )T .

 

If you can ?nd a region U0 of the plane R2 , such that: a) the expression of the function

 

min{|x|,|y|}

 

max{|x|,|y|} is particularly simple when (x, y) ? U0 , and b) there exist regions U1 , . . . , Un?1

 

such that, for all i, Ui is obtained as a rotation of U0 , and {U0 , U1 , . . . , Un?1 } is a partition

 

of the whole plane R2 , then you can solve the problem by justifying the following steps:

 

? Show that you can write E min{|X|, |Y |}

 

=n

 

max{|X|, |Y |} U0 ? Use the particularly simple expression of

 

calculate the integral above. min{|x|, |y|} 1 ?(x2 +y2 )/2

 

e

 

dxdy

 

max{|x|, |y|} 2?

 

min{|x|,|y|}

 

max{|x|,|y|} valid in the region U0 , in order to Of course, the ability consists of ?nding such region U0 and the number n of rotations of such

 

region which form a partition of the plane.

 


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