# Finding bias of the coin from noise corrupted signals

• I
• Amitkumarr
In summary: But with 2200 samples, you could also estimate the population mean and variance of the noise using standard formulas. So the bias of the coin is not really needed.In summary, A and B have personal communication systems and B has a biased coin. A asks B to toss the coin 2000 times and send the results, which are corrupted by noise. A also requests 200 zeros be sent before the coin toss results. From the 200 zeros, the mean and variance of the noise can be determined. To find the bias of the coin, the mean of the 2000 flips can be used. However, it may be more useful to use the mean and variance of the noise from the 200 zeros to estimate the population
Amitkumarr
TL;DR Summary
I have been given a dataset of noise corrupted sample with noise having a gaussian PDF. How do I find the bias of the coin in the given problem statement ?
Suppose there are two persons A and B such that both have a personal communication system which can transmit and receive bits. B has a biased coin whose bias is not known. A asks B to toss the coin 2000 times, send a 0 when a tail comes up and a 1 when a head comes up. It is known that whatever A receives is corrupted by noise, which has a Gaussian PDF with mean μ and variance σ2 . A put’s an additional request to B and asks B to simply send 200 zeros before sending the coin toss results. Using these 2200 samples of data, find the mean, variance of noise and also the bias of the coin.

My attempt:

From the 200 zeros that are sent first, we can determine the noise parameters like mean and variance because N + 0 = N , where N is the noise. But how do I find the bias of the coin from the remaining 2000 samples ?

Can anyone help me with the right approach to this problem ?

The effect of the noise is unknown. since the test did not show what happens when 200 1's are sent. My guess use the mean of the 2000 flips to determine bias.

Amitkumarr
mathman said:
The effect of the noise is unknown. since the test did not show what happens when 200 1's are sent. My guess use the mean of the 2000 flips to determine bias.
How do I determine bias from the mean of the 2000 flips ? I know that we can take Y=N+S where Y is the signal received at B including the noise(N) and S is the original signal sent by A. Noise is assumed to be additive and I know mean and variance of N and Y. Then how should I proceed ?

Amitkumarr
Amitkumarr said:
How do I determine bias from the mean of the 2000 flips ? I know that we can take Y=N+S where Y is the signal received at B including the noise(N) and S is the original signal sent by A. Noise is assumed to be additive and I know mean and variance of N and Y. Then how should I proceed ?
I think the bias would be the mean minus 1000, which is the number of heads/tails in a fair coin.

WWGD said:
I think the bias would be the mean minus 1000, which is the number of heads/tails in a fair coin.
Thanks for the reply. I have one small doubt:- If Z=X + Y where X is a continuous random variable and Y is a discrete random variable then is E[Z]=E[X] + E[Y] true for this case ? where E[Z] is the expectation value of Z.

Amitkumarr said:
Thanks for the reply. I have one small doubt:- If Z=X + Y where X is a continuous random variable and Y is a discrete random variable then is E[Z]=E[X] + E[Y] true for this case ? where E[Z] is the expectation value of Z.
Yes, expectation is always linear( as long as individual expectations exist, of course): Expectation of sum is the sum of the expectations.

Amitkumarr
WWGD said:
Yes, expectation is always linear( as long as individual expectations exist, of course): Expectation of sum is the sum of the expectations.
Thanks, now I have got an idea of how to solve this problem.

WWGD
Amitkumarr said:
Thanks, now I have got an idea of how to solve this problem.
Just curious, are you using H/N=H/2000 : Number of heads ( equiv., tails) as an estimator?

It appears to me that you are trying to find the bias of the noise. You can't know that from what you are given. I think they are assuming the noise in unbiased. It is safe for you to assume that too because otherwise you can't get the answer. If you really want to protect yourself you can make a note of this in your answer.

Usually in statistics you can't get a pure sample of the noise, so seldom could you measure the bias of the noise. Since you don't know what it is, you just assume it is zero. It's the best you can do.

The vocabulary of the discussion should be corrected! One cannot "find" population parameters only using sample data. The correct terminology is "estimate". And the discussion involves "finding" the values of commonly used estimators.

Stephen Tashi said:
The vocabulary of the discussion should be corrected! One cannot "find" population parameters only using sample data. The correct terminology is "estimate". And the discussion involves "finding" the values of commonly used estimators.
I guess you could estimate the population bias using the sample of 200 A's sent with Number of 1s received : ## \frac {Total 1s}{200}-1/2##

## 1. How do you define "bias" in the context of coin flipping?

In the context of coin flipping, bias refers to the tendency of a coin to land on one side more often than the other. This can be due to various factors such as the weight distribution of the coin, the force with which it is flipped, or external influences.

## 2. What is meant by "noise corrupted signals" in relation to finding bias of a coin?

Noise corrupted signals refer to the interference or random fluctuations in the data collected from coin flips. This could be caused by factors such as air resistance, uneven surfaces, or human error. These noise signals can make it difficult to accurately determine the bias of a coin.

## 3. How do you determine the bias of a coin from noise corrupted signals?

To determine the bias of a coin from noise corrupted signals, multiple coin flips must be performed and the results analyzed. By recording the frequency of heads and tails, statistical analysis can be used to identify any patterns or biases in the data.

## 4. Can a biased coin produce noise corrupted signals?

Yes, a biased coin can still produce noise corrupted signals. While the bias may affect the overall outcome of the coin flips, there may still be random fluctuations in the data due to external factors. This is why multiple flips are necessary to accurately determine the bias of a coin.

## 5. How can the bias of a coin impact the results of a study or experiment?

The bias of a coin can significantly impact the results of a study or experiment. If a coin is biased towards one side, it can skew the data and lead to inaccurate conclusions. This is why it is important to identify and account for any bias in coin flipping experiments.

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