Consider the random variables X, Y and Z, which are given to be pairwise uncorrelated (i.e., X and Y are uncorrelated, X and Z are uncorrelated, and Y and Z are uncorrelated). Suppose that
E[X]=E[Y]=E[Z]=0,
E[X2]=E[Y2]=E[Z2]=1,
E[X3]=E[Y3]=E[Z3]=0,
E[X4]=E[Y4]=E[Z4]=3.
Let W=a+bX+cX2 and V=dX, where a,b,c, and d are constants, all greater than 0.
Find the correlation coefficients ρ(X−Y,X+Y), ρ(X+Y,Y+Z), ρ(X,Y+Z) and ρ(W,V).
ρ(X−Y,X+Y)=- unanswered
ρ(X+Y,Y+Z)=- unanswered
ρ(X,Y+Z)=- unanswered
ρ(X−Y,X+Y)= 0
ρ(X+Y,Y+Z)= 0.5
ρ(X,Y+Z)= 0
ρ(W,V)= (b)/((b^2+2c^2)^0.5)
hope it helps! I am needing the other answers to the whole problem set
Oh, correlation coefficients? You're getting into some serious math territory there. Let me put on my clown glasses and give it a shot!
To calculate the correlation coefficients, we need to first find the covariance between the given random variables and then divide it by the product of their standard deviations. But don't worry, I'll crunch those numbers for you!
Let's start with ρ(X−Y,X+Y). Since X and Y are uncorrelated, their covariance is zero. We also know that E[X] = E[Y] = 0. Therefore, the correlation coefficient is also zero. So ρ(X−Y,X+Y) = 0.
Now, let's move on to ρ(X+Y,Y+Z). Again, X and Y are uncorrelated, so their covariance is zero. Similarly, Y and Z are uncorrelated, so their covariance is also zero. Since the covariance between (X+Y) and (Y+Z) is the sum of these covariances, it is also zero. Therefore, ρ(X+Y,Y+Z) = 0.
Next up, ρ(X,Y+Z). X and Y are uncorrelated, and Y and Z are uncorrelated as well. But that doesn't necessarily mean that X and Z are uncorrelated. However, we know that the correlation coefficient is symmetric. So we can say ρ(X,Y+Z) = ρ(Y+Z,X).
Finally, let's tackle ρ(W,V). W is a function of X and V is a function of X as well. Recall that X and V are uncorrelated, so their covariance is zero. It means that the correlation coefficient between W and V is also zero. In other words, ρ(W,V) = 0 (cue the sad clown music).
So, in summary:
ρ(X−Y,X+Y) = 0
ρ(X+Y,Y+Z) = 0
ρ(X,Y+Z) = ρ(Y+Z,X) (we can't determine the exact value without more information)
ρ(W,V) = 0
Math is serious business, but I hope I managed to lighten it up a little!
To find the correlation coefficients ρ(X−Y,X+Y), ρ(X+Y,Y+Z), ρ(X,Y+Z), and ρ(W,V), we need to compute the covariances and variances of the respective random variables.
First, let's find the covariance and variances of X and Y:
Cov(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY]
(Since E[X] and E[Y] are both 0)
= E[XY] = 0
Var(X) = E[(X - E[X])^2] = E[X^2] = 1
Var(Y) = E[(Y - E[Y])^2] = E[Y^2] = 1
Similarly, we find the covariance and variances of X and Z:
Cov(X, Z) = E[(X - E[X])(Z - E[Z])] = E[XZ]
= E[XZ] = 0
Var(Z) = E[(Z - E[Z])^2] = E[Z^2] = 1
Finally, we find the covariance and variances of Y and Z:
Cov(Y, Z) = E[(Y - E[Y])(Z - E[Z])] = E[YZ]
= E[YZ] = 0
Var(Y) = E[(Y - E[Y])^2] = E[Y^2] = 1
Now, let's calculate the correlation coefficients:
1. ρ(X−Y,X+Y):
Cov(X - Y, X + Y) = Cov(X, X) - Cov(X, Y) + Cov(-Y, X) - Cov(-Y, Y)
= Var(X) - Cov(X, Y) - Cov(Y, X) + Var(Y)
= 1 - 0 - 0 + 1 = 2
ρ(X−Y,X+Y) = Cov(X - Y, X + Y) / sqrt(Var(X - Y) * Var(X + Y))
= 2 / sqrt(Var(X - Y) * Var(X + Y))
2. ρ(X+Y,Y+Z):
Cov(X + Y, Y + Z) = Cov(X, Y) + Cov(X, Z) + Cov(Y, Y) + Cov(Y, Z)
= 0 + 0 + 1 + 0 = 1
ρ(X+Y,Y+Z) = Cov(X + Y, Y + Z) / sqrt(Var(X + Y) * Var(Y + Z))
= 1 / sqrt(Var(X + Y) * Var(Y + Z))
3. ρ(X,Y+Z):
Cov(X, Y + Z) = Cov(X, Y) + Cov(X, Z)
= 0 + 0 = 0
ρ(X,Y+Z) = Cov(X, Y + Z) / sqrt(Var(X) * Var(Y + Z))
= 0 / sqrt(Var(X) * Var(Y + Z)) = 0
4. ρ(W,V):
To find ρ(W,V), we first need to find Cov(W, V) and Var(W):
Cov(W, V) = Cov(a + bX + cX^2, dX)
= bcov(X^2, X) + acov(X, X) + dcov(X^2, X)
= bE[X^3] + aE[X^2] + dE[X^3]
= b(0) + a(1) + d(0)
= a
Var(W) = Var(a + bX + cX^2)
= Var(bX + cX^2)
= b^2Var(X) + c^2Var(X^2)
= b^2 + c^2
ρ(W,V) = Cov(W, V) / sqrt(Var(W) * Var(V))
= a / sqrt((b^2 + c^2) * Var(X))
Please provide the values of a, b, and c in order to calculate ρ(W,V).