Let X and Y be independent random variables with zero means, and variances 1 and 2, respectively. Let U=X+Y and V=X+2Y .

Find the coefficients a and b of the Linear Least Mean Squares (LLMS) estimator VˆL=aU+b of V based on U .

find a and b.

a=5/3

b=0

Are you Sure of 5/3?

Nope..

But as Vllms = E[V]+cov(U,V)/var(U)*(U-E[U]) i can't find a mistake in my calculations..

and that was the right answers...

Why did the random variables go to therapy? Because they had zero means and suffered from a case of confusion! But don't worry, I'm here to help them out and find those coefficients!

To find the coefficients a and b of the Linear Least Mean Squares (LLMS) estimator V^L = aU + b of V based on U, we can use the properties of covariance.

Since X and Y are independent, the covariance between X and Y is zero, Cov(X, Y) = 0.

Now, let's calculate the covariance between U and V:

Cov(U, V) = Cov(X + Y, X + 2Y)
= Cov(X, X) + Cov(X, 2Y) + Cov(Y, X) + Cov(Y, 2Y)
= Var(X) + 2Cov(X, Y) + Cov(Y, 2Y)
= Var(X) + Cov(Y, 2Y)
= Var(X) + 2Var(Y) (since Cov(X, Y) = Cov(Y, X) = 0)

Since Var(X) = 1 and Var(Y) = 2, we have:
Cov(U, V) = 1 + 2(2) = 5

Next, let's calculate the variance of U:

Var(U) = Var(X + Y)
= Var(X) + Var(Y) + 2Cov(X, Y)
= Var(X) + Var(Y) (since Cov(X, Y) = Cov(Y, X) = 0)
= 1 + 2 (since Var(X) = 1 and Var(Y) = 2)
= 3

Now, we can calculate the coefficient a using the formula for the LLMS estimator:

a = Cov(U, V) / Var(U)
= 5 / 3

And finally, we can calculate the coefficient b:

b = E(V) - aE(U) (E denotes expectation or mean)
= E(X + 2Y) - aE(X + Y)
= E(X) + 2E(Y) - aE(X) - aE(Y)
= 0 + 2(0) - a(0) - a(0)
= 0

So, the coefficients for the LLMS estimator V^L = aU + b of V based on U are:
a = 5/3
b = 0

Ta-da! The coefficients have been found. Now, go forth and estimate away!

To find the coefficients a and b of the Linear Least Mean Squares (LLMS) estimator VˆL=aU+b of V based on U, we need to minimize the mean squared error (MSE) between V and VˆL.

The first step is to find the covariance matrix of the random variables U and V. Since X and Y are independent random variables, we have:

Cov(X, Y) = 0
Cov(U, V) = Cov(X + Y, X + 2Y) = Cov(X, X) + Cov(X, 2Y) + Cov(Y, X) + Cov(Y, 2Y)
= Var(X) + 2Cov(X, Y) + Cov(Y, Y) + 2Var(Y) = Var(X) + 2Cov(Y, Y) + 2Var(Y)
= 1 + 2 * 0 + 2 * 2 = 5

Next, we need to find the variance of U. Since X and Y are independent and have variances 1 and 2 respectively, we have:

Var(U) = Var(X + Y) = Var(X) + Var(Y) = 1 + 2 = 3

Now, let's calculate the coefficients a and b using the LLMS formula:

a = Cov(U, V) / Var(U) = 5 / 3
b = E(V) - a * E(U)

Since X and Y have zero means, we have:

E(U) = E(X + Y) = E(X) + E(Y) = 0 + 0 = 0
E(V) = E(X + 2Y) = E(X) + 2E(Y) = 0 + 2 * 0 = 0

Therefore, b = 0 - a * 0 = 0

In conclusion, to find the coefficients a and b of the LLMS estimator VˆL=aU+b of V based on U, we have a = 5/3 and b = 0.