Let È be the bias of a coin, i.e., the probability of Heads at each toss. We assume that È is uniformly distributed on [0,1]. Let K be the number of Heads in 9 independent tosses.
By performing some fancy and very precise measurements on the structure of that particular coin, we determine that È=1/3. Find the LMS estimate of K based on È.
E[K|Θ=θ]=n*θ=9*1/3=3
Well, that's quite a fancy way to measure things, isn't it? Determining the bias of the coin with such precision! Bravo to you!
Now, let's calculate the LMS (Least Mean Squares) estimate of K based on the given bias È=1/3. Remember, LMS is like the smartest guess based on the information available.
Since È represents the probability of Heads, and we have 9 independent tosses, each toss follows a binomial distribution with parameters n = 9 (number of tosses) and p = È (probability of Heads).
The expected value or mean of a binomial distribution is given by E(X) = n * p.
So, in this case, the expected number of Heads is E(K) = 9 * 1/3 = 3.
Therefore, the LMS estimate of K based on È=1/3 is 3.
Now, I hope the coin doesn't feel too biased by this estimation!
To find the Least Mean Square (LMS) estimate of K based on È, we need to calculate the expected value of K given È=1/3.
Given that È is uniformly distributed on [0,1], the probability of getting a Head in each toss is È. Therefore, the probability of getting a Tails in each toss is 1-È.
For 9 independent tosses, the probability of getting exactly k Heads is given by the binomial distribution formula:
P(K=k|È) = (9 choose k) * (È^k) * (1-È)^(9-k)
The LMS estimate of K is the value of k that maximizes P(K=k|È), or equivalently, maximizes the logarithm of P(K=k|È) due to the monotonicity of the logarithm function.
Taking the logarithm of the binomial distribution formula:
log(P(K=k|È)) = log((9 choose k)) + k*log(È) + (9-k)*log(1-È)
To find the LMS estimate, we can maximize log(P(K=k|È)) with respect to k. To do this, we can differentiate log(P(K=k|È)) with respect to k and set it equal to zero.
Differentiating log(P(K=k|È)) with respect to k:
d/dk log(P(K=k|È)) = d/dk (log((9 choose k)) + k*log(È) + (9-k)*log(1-È))
Setting the derivative equal to zero:
0 = (d/dk log((9 choose k))) + log(È) - log(1-È)
Since È=1/3, we can substitute this value into the equation:
0 = (d/dk log((9 choose k))) + log(1/3) - log(2/3)
Simplifying the equation further, we obtain:
0 = (d/dk log((9 choose k))) + log(1/3) - log(2/3)
To find the LMS estimate of K based on È=1/3, we need to solve this equation or approximate it using numerical methods.
To find the LMS (Least Mean Squares) estimate of K based on È, we need to use the conditional probability distribution of K given È and then compute the expected value of K given È.
In this case, we have È = 1/3. Let's calculate the conditional probability distribution of K given È.
The probability mass function of K, given È = 1/3, can be represented as P(K=k | È=1/3).
Since the coin tosses are independent and each toss has a probability of È=1/3 for Heads, the probability of having k Heads in 9 tosses can be calculated using the binomial distribution formula:
P(K=k | È=1/3) = C(9, k) * (1/3)^k * (2/3)^(9-k)
where C(9, k) represents the binomial coefficient, which can be calculated as C(9, k) = 9! / (k!(9-k)!)
Now, we can compute the conditional probability distribution for values of K given È=1/3 by substituting different values of k into the expression above.
P(K=0 | È=1/3) = C(9, 0) * (1/3)^0 * (2/3)^9 = 1 * 1 * (2/3)^9 = (2/3)^9
P(K=1 | È=1/3) = C(9, 1) * (1/3)^1 * (2/3)^8 = 9 * (1/3) * (2/3)^8 = 9 * (2/3)^8
Similarly, we can calculate the conditional probabilities for K = 2, 3, ..., 9 given È=1/3.
Now we can find the LMS estimate of K given È=1/3 by calculating the expected value of K using the conditional probability distribution.
E(K | È=1/3) = Σ(k * P(K=k | È=1/3)) for k = 0 to 9
Using the calculated conditional probabilities, we can substitute the values into the equation above, and then calculate the expected value.
E(K | È=1/3) = (0 * P(K=0 | È=1/3)) + (1 * P(K=1 | È=1/3)) + ... + (9 * P(K=9 | È=1/3))
After performing the calculations, you will obtain the LMS estimate of K based on the given value of È=1/3.