Q2. Electrical engineers are considering two altenative systems for sending messages. A message consists of a word either a 0 or a 1 However, because of random noise in the channel, al that is transmitted could be received as 0 and vice versa. That is, there is a small probability q that P(A transmitted 1 is received as 0) = P(A transmitted 0 is received as 1) = q One scheme is to send a single digit. A second scheme is to repeat the selected digit 3 times in succession. At the receiving end, the majority rule be used to decode. For instance, when any of 101. 110 or 011 are received, it is interpreted to mean a 1 was sent. (a) Evaluate the probability that a transmitted 1 will be received as a 1 under the 3-digit q = 0.05. Compare with scheme when a single digit is transmitted as a scheme when word. Treat the result for different digits as independent. (b) Suppose a message, consisting of the two words, a 1 followed by a 0, is to transmitted using 3-digit scheme. What is the probability that total message will be correctly decoded under the majority nule with q = 0.05?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Given,
Electrical engineers are considering two alternative systems for sending messages.
A message consists of a word either a 0 or a 1.
That is, there is a small probability ? that ?(? transmitted 1 is received as 0) = ?(? transmitted 0 is received as 1) = ?
One scheme is to send a single digit.
A second scheme is to repeat the selected digit 3 times in succession.
At the receiving end, the majority rule be used to decode.
(a)
Let X be the number of incorrectly decoded digits.
Then X ~ Binomial(n = 3, p = 0.05)
For majority rule, X should be at most 1 to correctly decode the digit.
Probability that a transmitted 1 will be received as a 1 under the 3-digit scheme = P(X 1)
= P(X = 0) + P(X = 1)
= 3C0(0.05)0(1-0.05)3-0 + 3C1(0.05)1(1-0.05)3-1
= 0.99275
Probability of correctly decoding a digit with scheme when a single digit is transmitted as a word = 1 - q
= 1 - 0.05 = 0.95
Thus the Probability of correctly decoding a digit with scheme when a single digit is transmitted is less than with 3-digit scheme.
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