Let X,Y,Z be independent discrete random variables with E[X]=2, E[Y]=0, E[Z]=0, E[X^2]=20, E[Y^2]=E[Y^2]=16, and Var(X)=Var(Y)=Var(Z)= 16. Let A=X(Y+Z) and B=XY.
1.Find E[B].
2.Find Var(B).
3.Find E[AB].
4. are A and B independent?
5.Are A and B are conditionally independent, given X=0.
6.Are A and B are conditionally independent, given X=1.
1. 0
2. 320
3. 320
correct me if am wrong.
but if you solve it from this method:
E[AB]=E[{x(y+z)}*{xy}]=E[x^2.y^2+x^2.y.z] = E[x^2].E[y^2] + E[x^2].E[y].E[z]
= 20.16+0=320.
is this wrong?
1. 0
2. 320
3. 0
E[A*B] = E[A] * E[B]
= E[X(Y+Z)] * E[X*Y]
= (E[X*Y + Y*Z]) * E[X]*E[Y]
= (E[X]*E[Y] + E[Y]*E[Z]) * E[X]*E[Y] since E[X]=2,E[Y]=0
= ((2 * 0) + (0 * 0) ) * 2*0 since E[Y]=0,E[Z]=0
E[AB] = 0
Also, do you know the answer for 4,5, and 6?
1. E[B]:
Since X and Y are independent, we can use the formula E[XY] = E[X] * E[Y]:
E[B] = E[XY] = E[X] * E[Y] = 2 * 0 = 0.
2. Var(B):
Since X and Y are independent, we can use the formula Var(B) = Var(X) * E[Y^2] + Var(Y) * E[X^2] + Var(X) * Var(Y):
Var(B) = Var(X) * E[Y^2] + Var(Y) * E[X^2] + Var(X) * Var(Y) = 16 * 16 + 16 * 20 + 16 * 16 = 512.
3. E[AB]:
Since A = X(Y + Z), we have:
E[AB] = E[X(Y + Z) * XY] = E[X^2 * Y^2 + XYZ^2 + XY^2Z] = E[X^2] * E[Y^2] + E[X] * E[Y] * E[Z^2] + E[X] * E[Y^2] * E[Z]
= 20 * 16 + 2 * 0 * E[Z^2] + 2 * 16 * E[Z] = 320 + 0 + 32 * E[Z].
4. Are A and B independent?
To determine if A and B are independent, we need to check if P(A∩B) = P(A) * P(B).
Since A = X(Y + Z) and B = XY, we can see that A and B are not independent since A contains the term Z, which is not present in B.
5. Are A and B conditionally independent, given X = 0?
To determine if A and B are conditionally independent given X = 0, we need to check if P(A∩B|X = 0) = P(A|X = 0) * P(B|X = 0).
If X = 0, then both A and B will be 0. Therefore, A and B are conditionally independent given X = 0.
6. Are A and B conditionally independent, given X = 1?
To determine if A and B are conditionally independent given X = 1, we need to check if P(A∩B|X = 1) = P(A|X = 1) * P(B|X = 1).
If X = 1, then A = Y + Z and B = Y. Since A contains the term Z, which is not present in B, A and B are not conditionally independent given X = 1.
To solve these questions, we need to use some key properties of expectation and variance. Let's tackle each question step by step:
1. Finding E[B]:
The expectation of a product of independent random variables is equal to the product of their individual expectations. Since X and Y are independent, we can write:
E[B] = E[XY] = E[X]E[Y]
Given that E[X] = 2 and E[Y] = 0, we can substitute these values into the equation:
E[B] = 2 * 0 = 0
Therefore, E[B] = 0.
2. Finding Var(B):
The variance of a product of independent random variables is not simply the product of their individual variances. However, we can use the covariance to find the variance of B.
Covariance of X and Y can be calculated as:
Cov(X, Y) = E[XY] - E[X]E[Y]
We have already found that E[XY] = 0, E[X] = 2, and E[Y] = 0, so:
Cov(X, Y) = 0 - (2 * 0) = 0
Since X and Y are independent, Cov(X, Y) = 0 means they are uncorrelated.
Next, we can use the formula for Var(B) in terms of the covariance:
Var(B) = E[B^2] - E[B]^2
For B = XY:
E[B^2] = E[(XY)^2] = E[X^2Y^2] = E[X^2]E[Y^2]
Given that E[X^2] = 20 and E[Y^2] = 16, we can substitute these values into the equation:
E[B^2] = 20 * 16 = 320
We already found that E[B] = 0, so we can calculate the variance:
Var(B) = E[B^2] - E[B]^2 = 320 - 0^2 = 320
Therefore, Var(B) = 320.
3. Finding E[AB]:
To find E[AB], we need to calculate the expectation of the product of A and B.
E[AB] = E[X(Y+Z)XY] = E[X^2Y^2 + X^2YZ]
Using the property that X and Y are independent, and Y and Z are independent, we get:
E[AB] = E[X^2]E[Y^2] + E[X^2]E[Y]E[Z]
We already found that E[X^2] = 20, E[Y^2] = 16, E[Y] = 0, and E[Z] = 0, so substituting these values gives:
E[AB] = 20 * 16 + 20 * 0 * 0 = 320
Therefore, E[AB] = 320.
4. Are A and B independent?
To determine whether A and B are independent, we need to check if their joint distribution is equal to the product of their marginal distributions.
The joint distribution of A and B is unknown based on the information given, so we cannot definitively determine if they are independent.
5. Are A and B conditionally independent, given X = 0?
To determine conditional independence, we need to calculate the conditional distributions and check if they factorize.
In this case, given X = 0, A will always be equal to 0, regardless of the values of Y and Z. On the other hand, B will also be equal to 0, since anything multiplied by 0 is always 0.
Since both A and B are always zero when X = 0, they are trivially conditionally independent.
6. Are A and B conditionally independent, given X = 1?
Similar to the previous case, given X = 1, A will be equal to Y + Z, and B will be equal to Y.
The conditional distribution of A, given X = 1, is dependent on both Y and Z. Similarly, the conditional distribution of B, given X = 1, is dependent on Y.
Since the conditional distributions of A and B are dependent on the same random variable, Y, they are not conditionally independent given X = 1.
Note: It is not possible to determine the exact conditional distributions without additional information about the joint distribution of X, Y, and Z.