Question

A binary message m, where m is equal either to 0 or to 1, is sent over an information channel. Assume that if m = 0, the value s = −1.5 is sent, and if m = 1, the value s = 1.5 is sent. The value received is X, where X = s + E, and E ∼ N(0, 0.66). If X ≤ 0.5, then the receiver concludes that m = 0, and if X > 0.5, then the receiver concludes that m = 1.

a) If the true message is m = 0, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 1? Round the answer to four decimal places.

b) If the true message is m = 1, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 0?

c) A string consisting of 60 1s and 40 0s will be sent. A bit is chosen at random from this string. What is the probability that it will be received correctly?

d) Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 1, what is the probability that the bit sent was 0?

e) Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 0, what is the probability that the bit sent was 1?

Answer #1

A binary message m, where m is equal either to 0 or to 1, is
sent over an information channel. Assume that if m = 0, the value s
= −1.5 is sent, and if m = 1, the value s = 1.5 is sent. The value
received is X, where X = s + E, and E ∼ N(0, 0.66). If X ≤ 0.5,
then the receiver concludes that m = 0, and if X > 0.5, then the...

A
binary message m, where m is equal either to 0 or to 1, is sent
over an information channel. Because of noise in the channel, the
message received is X, where X = m + E, and E is a random variable
representing the channel noise. Assume that if X ≤ 0.5 then the
receiver concludes that m = 0 and that if X > 0.5 then the
receiver concludes that m = 1. Assume that E ∼ N(0,...

1. A transmitted message bit B is either 1 or 0, each with equal
likelihood.
The bit is captured by N independent receivers; each receiver
has a
probability p of receiving the wrong bit value (e.g. receiving a
1 when
a 0 was sent).
Write the conditional probability that the message bit is 1, if
all N
receivers receive a 1. Write your answer as a function of p.

In a binary communication system, 1’s are sent twice as
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correct decision as to which it was only 3/4 of the time. Errors in
di↵erent symbol transmissions are independent.
(a) Suppose that the string of symbols 1001 is transmitted. What
is the probability that all symbols in the string are received
correctly?
(b) Find the probability of an error being incurred as a result
of the receiver making the...

(1) Consider a noisy wireless channel which flips each bit ( 0
becomes 1 or 1 becomes 0 ) with probability p. To guarantee the
information is received reliably, each bit is transmitted m times
(called repetition coding). For example, if m = 3 and “0” needs to
be sent, you transmit “000” instead of “0”.
(1)
Consideranoisywirelesschannelwhichflipseachbit(0becomes1or1becomes0)withprobabilityp.
To guarantee the information is received reliably, each bit is
transmitted m times (called repetition coding). For example, if m =
3...

In a binary communication channel, 0s and 1s are transmitted
with equal probability. The probability that a 0 is correctly
received (as a 0) is 0.99. The probability that a 1 is correctly
received (as a 1) is 0.90. Suppose we receive a 1, what is the
probability that, in fact, a 0 was sent?
How to apply bayes rule?

A binary communication channel transmits a sequence of "bits"
(0s and 1s). Suppose that for any particular bit transmitted, there
is a 15% chance of a transmission error (a 0 becoming a 1 or a 1
becoming a 0). Assume that bit errors occur independently of one
another. (Round your answers to four decimal places.)
(a)
Consider transmitting 1000 bits. What is the approximate
probability that at most 165 transmission errors occur?
(b)
Suppose the same 1000-bit message is sent...

Consider a noisy communication channel, where each bit is
flipped with probability p (the probability that a bit is sent in
error is p). Assume that n−1 bits, b1,b2,⋯,b(n−1), are going to be
sent on this channel. A parity check bit is added to these bits so
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can distinguish occurrence of odd number of errors, that is, if
one, three, or any odd number of errors occur, the...

The receiver in an optical communications system uses a
photodetector that counts the number of photons that arrive during
one time unit. Suppose that the number X of photons can be modeled
as a Poisson random variable with rate λ1 when a signal is present
(say bit “1” is transmitted) and a Poisson random variable with
rate λ0 < λ1 when a signal is absent (say bit “0” is
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3. Each time a modem transmits one bit, the receiving modem
analyzes the signal arrives and decides whether the transmitted bit
is 0 or 1. It makes an error with probability p,
independent of whether any other bit is received correctly. a) If
the transmission continues until first error, what is the
distribution of random variable X, the number of bits transmitted?
b) If p = 0.1, what is probability that X = 10? c) what is
probability that X...

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