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[Z^2]=16,
and
Var(X)=Var(Y)=Var(Z)=16.
Let A=X(Y+Z) and B=XY.
1. Find E[B].
E[B]= ?

2. Find Var(B).
Var(B)= ?

3. Find E[AB].
E[AB]= ?

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1. Find E[B].

E[B] = E[XY]

Since X and Y are independent, E[XY] = E[X] * E[Y]

Given that E[X] = 2 and E[Y] = 0, we can calculate:

E[B] = 2 * 0 = 0

Therefore, E[B] = 0.

2. Find Var(B).
Var(B) = Var(XY)

Since X and Y are independent, Var(XY) = Var(X) * Var(Y)

Given that Var(X) = Var(Y) = 16, we can calculate:

Var(B) = 16 * 16 = 256

Therefore, Var(B) = 256.

3. Find E[AB].
E[AB] = E[X(Y+Z)]

Since X, Y, and Z are independent, E[X(Y+Z)] = E[X] * E[Y+Z]

Given that E[X] = 2, E[Y] = 0, and E[Z] = 0, we can calculate:

E[AB] = 2 * (0 + 0) = 2 * 0 = 0

Therefore, E[AB] = 0.

To find the expected value (E[B]) of a random variable B, we need to calculate the sum of the products of B and its corresponding probabilities.

1. First, let's find the probability distribution of B by finding all possible values of B and their corresponding probabilities.

Since X, Y, and Z are independent random variables, the probability distribution of B can be obtained as follows:

B = XY

For each possible value of X and Y, we need to multiply them to find the corresponding value of B.

Possible values of X: {2}
Possible values of Y: {0}

Since X and Y take only one value each, the probability of each combination is 1.

So, for B = XY, there is only one possibility: B = 2 * 0 = 0.

Therefore, the probability distribution of B is as follows:

B -> P(B)
0 -> 1

Now, we can calculate the expected value E[B]:

E[B] = Σ B * P(B)

In this case, there is only one possible value of B, which is 0, with a probability of 1.

E[B] = 0 * 1 = 0

So, E[B] = 0.

2. To find the variance Var(B) of a random variable B, we need to calculate E[(B - E[B])^2].

First, we need to calculate E[B] (which we already found to be 0 in step 1).

Second, we need to calculate E[B^2]:

Since B = XY,

B^2 = (XY)^2 = X^2Y^2

Since X and Y are independent variables, we can calculate E[B^2] as follows:

E[B^2] = E[X^2Y^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:

E[B^2] = 20 * 16 = 320

Now, we can calculate Var(B):

Var(B) = E[B^2] - (E[B])^2

Var(B) = 320 - (0)^2

Var(B) = 320

So, Var(B) = 320.

3. To find the expected value E[AB], we need to calculate the sum of the products of AB and their corresponding probabilities.

Since A = X(Y + Z), we can calculate E[AB] as follows:

E[AB] = E[X(Y + Z)B]

The random variables X and B are independent, so we can split the expectations:

E[AB] = E[X(Y + Z)] * E[B]

Given that E[X] = 2, E[Y + Z] = E[Y] + E[Z] = 0 + 0 = 0 (since Y and Z are independent), and E[B] = 0 (as found in step 1):

E[AB] = 2 * 0 = 0

So, E[AB] = 0.