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Refer to Exercise 23. Assume that if m = 0, the value s =
Chapter 4, Problem 24E(choose chapter or problem)
Refer to Exercise 23. 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, 0.64). If \(X \leq 0.5\), 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?
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?
Equation Transcription:
Text Transcription:
X
X leq 0.5
Questions & Answers
QUESTION:
Refer to Exercise 23. 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, 0.64). If \(X \leq 0.5\), 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?
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?
Equation Transcription:
Text Transcription:
X
X leq 0.5
ANSWER:Solution 24E
Step1 of 6:
We have a random variable X it presents the number of messages received.
Let us assume that if m = 0 then the value of s = -1.5
if m = 1 then the value of s = 1.5
Let X = s + E
Where,
E = a random variable representing the channel noise.
EN(0, 0.64)
m = Binary message.
If X 0.5 then the receiver concludes that m = 0
If X > 0.5 then the receiver concludes that m = 1
Here our goal is:
a).We need to find the probability of an error when the true message is m = 0.
b).We need to find the probability of an error when the true message is m = 1.
c).We need to find the probability that it will be received correctly when a string consisting of 60 1s and 40 0s will be sent.
d).We need to find the probability that the bit sent was 0?
e).We need to find the probability that the bit sent was 1?
Step2 of 6:
a).
We have
X = s + E
Above equation can also written as
E = X - s
= 0.5 - (-1.5) [therefore, s = -1.5 when m = 0]
= 2
The probability of an error is given by
P(X > 0.5) = P(Z > )
= 1 - P(Z < )
= 1 - P(Z < )
= 1 - P(Z < 3.125)
Here, P(Z < 3.125) is obtained from standard normal table(area under normal curve)
P(Z < 3.125) = 0.9991 (In standard normal table we have to see in row 3.1 under column 0.02)
Now,
P(X > 0.5) = 1 - P(Z < 3.125)
= 1 - 0.9991
= 0.0009
Hence, P(X > 0.5) = 0.0009.
Step3 of 6: