3.) If the coin is flipped 6 times and the number of Heads obtained is 3, what is the resulting conditional mean squared error of the least mean squares (LMS) estimator for the bias of the coin?
To find the resulting conditional mean squared error (MSE) of the least mean squares (LMS) estimator for the bias of the coin, we need some additional information. Specifically, we need to know the assumed bias of the coin and the variance of the coin's outcomes.
The LMS estimator is a method used to estimate an unknown parameter by minimizing the mean squared error. In this case, we want to estimate the bias of the coin.
Let's assume that the bias of the coin is denoted by p, where p represents the probability of obtaining a Heads when a coin is flipped.
To find the LMS estimator for the bias of the coin, we calculate the conditional expectation of the bias given the observed data. In this case, the observed data is that the coin is flipped 6 times, and 3 of those flips result in Heads.
The conditional mean squared error (MSE) is defined as the mean of the squared difference between the estimated bias and the true bias.
Now, let's consider the steps to find the resulting conditional MSE of the LMS estimator:
Step 1: Calculate the LMS estimator, denoted by p̂, which is the conditional expectation of the bias given the observed data. In this case, p̂ is equal to the ratio of the number of observed Heads (3) to the total number of flips (6). So, p̂ = 3/6 = 0.5.
Step 2: Calculate the difference between the estimated bias (p̂) and the true bias (p), which is p̂ - p. Since we don't know the true bias (p) in this case, we can't determine this difference.
Step 3: Square the difference obtained in Step 2, which gives (p̂ - p)^2. Again, we can't calculate this term without knowing the true bias (p).
Step 4: Repeat Steps 2 and 3 multiple times and take the average of all the squared differences obtained to calculate the MSE of the LMS estimator. However, since we don't have the necessary information, we cannot proceed with this step.
In conclusion, without knowing the true bias (p) or the variance of the coin's outcomes, we cannot calculate the resulting conditional MSE of the LMS estimator for the bias of the coin.
To calculate the resulting conditional mean squared error of the least mean squares (LMS) estimator for the bias of the coin, we need to break down the problem into steps:
Step 1: Define the problem
We are given that a coin is flipped 6 times, and the number of Heads obtained is 3. We want to calculate the conditional mean squared error of the LMS estimator for the bias of the coin.
Step 2: Understand the LMS estimator for the bias
The LMS estimator for the bias of a coin is a statistical estimation technique that tries to estimate the bias, which is the probability of obtaining a Head (or a Tail) when the coin is flipped.
Step 3: Calculate the conditional mean squared error of the LMS estimator
To calculate the conditional mean squared error, we first need to know the bias estimator. In this case, since we are given that the number of Heads obtained is 3 out of 6 flips, we can estimate the bias as 3/6, which simplifies to 1/2.
Next, we need to calculate the variance of the bias estimator. The variance of the LMS estimator depends on the sample size and the true bias. In this case, the sample size is 6 (6 coin flips) and the true bias is 1/2.
Using the formula for variance of LMS estimator, which is (true bias) * (1 - true bias) / sample size, we can substitute the values:
Variance = (1/2) * (1 - 1/2) / 6 = 1/24.
Finally, to calculate the conditional mean squared error, we multiply the variance by the sample size again:
Conditional Mean Squared Error = Variance * Sample Size = (1/24) * 6 = 1/4.
Therefore, the resulting conditional mean squared error of the LMS estimator for the bias of the coin is 1/4.