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UCSD ECE
Prof. Tara Javidi Tuesday, April 13, 2021
Homework Set # Due: Thursday, April 22, 2021
- Time until the n-th arrival. Let the random variable N (t) be the number of packets arriving during time (0, t]. Suppose N (t) is Poisson with pmf
pN (n) =
(λt)n n!
e−λt^ for n = 0, 1 , 2 ,....
Let the random variable Y be the time to get the n-th packet. Find the pdf of Y.
- Diamond distribution. Consider the random variables X and Y with the joint pdf
fX,Y (x, y) =
c, if |x| + |y| ≤ 1 /
0 , otherwise,
where c is a constant.
(a) Find c. (b) Find fX (x) and fX|Y (x|y). (c) Are X and Y independent random variables? Justify your answer.
- First available teller. Consider a bank with two tellers. The service times for the tellers are independent exponentially distributed random variables X 1 ∼ Exp(λ 1 ) and X 2 ∼ Exp(λ 2 ), respectively. You arrive at the bank and find that both tellers are busy but that nobody else is waiting to be served. You are served by the first available teller once he/she is free. What is the probability that you are served by the first teller?
- Coin with random bias. You are given a coin but are not told what its bias (probability of heads) is. You are told instead that the bias is the outcome of a random variable P ∼ U[0, 1]. To get more infromation about the coin bias, you flip it independently 10 times. Let X be the number of heads you get. Thus X ∼ Binom(10, P ). Assuming that X = 9, find and sketch the a posteriori probability of P , i.e., fP |X (p|9).
- Optical communication channel. Let the signal input to an optical channel be given by
X =
λ 0 with probability (^12) λ 1 with probability 12.
The conditional pmf of the output of the channel Y |{X = λ 0 } ∼ Poisson(λ 0 ), i.e., Poisson with intensity λ 0 , and Y |{X = λ 1 } ∼ Poisson(λ 1 ). Show that the MAP rule reduces to
D(y) =
λ 0 , y < y∗ λ 1 , otherwise.
Let λ 0 = 1 and λ 1 = 2, find y∗^ and the corresponding probability of error. Compare with the case that λ 0 = 1 and λ 1 = 100.
- Iocane or Sennari. An absent-minded chemistry professor forgets to label two identi- cally looking bottles. One bottle contains a chemical named “Iocane” and the other bottle contains a chemical named “Sennari”. It is well known that the radioactivity level of “Iocane” has the U[0, 1] distribution, while the radioactivity level of “Sennari” has the Exp(1) distribution.
(a) Let X be the radioactivity level measured from one of the bottles. What is the optimal decision rule (based on the measurement X) that maximizes the chance of correctly identifying the content of the bottle? (b) What is the associated probability of error?
- Two independent uniform random variables. Let X and Y be independently and uni- formly drawn from the inverval [0, 1].
(a) Find the pdf of U = max(X, Y ). (b) Find the pdf of V = min(X, Y ). (c) Find the pdf of W = U − V. (d) Find the probability P{|X − Y | ≥ 1 / 2 }.
- Waiting time at the bank. Consider a bank with two tellers. The service times for the tellers are independent exponentially distributed random variables X 1 ∼ Exp(λ 1 ) and X 2 ∼ Exp(λ 2 ), respectively. You arrive at the bank and find that both tellers are busy but that nobody else is waiting to be served. You are served by the first available teller once he/she becomes free. Let the random variable Y denote your waiting time. Find the pdf of Y.
- Signal or no signal. Consider a communication system that is operated only from time to time. When the communication system is in the “normal” mode (denoted by M = 1), it transmits a random signal S = X with
X =
+1, with probability 1/ 2 , − 1 , with probability 1/ 2.
When the system is in the “idle” mode (denoted by M = 0), it does not transmit any signal (S = 0). Both normal and idle modes occur with equal probability. Thus
S =
X, with probability 1/ 2 , 0 , with probability 1/ 2.
The receiver observes Y = S +Z, where the ambient noise Z ∼ U[− 1 , 1] is independent of S.
(a) Find and sketch the conditional pdf fY |M (y|1) of the receiver observation Y given that the system is in the normal mode. (b) Find and sketch the conditional pdf fY |M (y|0) of the receiver observation Y given that the system is in the idle mode. (c) Find the optimal decoder D(y) for deciding whether the system is normal or idle. Provide the answer in terms of intervals of y. (d) Find the associated probability of error.
- Function of uniform random variables. Let X and Y be two independent U[0, 1] ran- dom variables. Find the probability density function (pdf) of Z = (X + Y ) mod 1 (i.e., Z = X + Y if X + Y ≤ 1 and X + Y − 1 if X + Y > 1).
- Maximal correlation.
(a) For any pair of random variables (X, Y ), show that
FX,Y (x, y) ≤ min{FX (x), FY (y)}.
Now let F and G be continuous and invertible cdf’s and let X ∼ F.
(b) Find the distribution of Y = G−^1 (F (X)).
(c) Show that FX,Y (x, y) = min{F (x), G(y)}.
