Let X1 , X2 , X3 be i.i.d. Binomial random variables with parameters n=2 and p=1/2 . Define two new random variables

Y1 =X1−X3,
Y2 =X2−X3.
We further introduce indicator random variables Zi∈{0,1} with Zi=1 if and only if Yi=0 for i=1,2 .

1. Calculate the covariance of Y1 and Y2.
(Give an exact answer or a decimal accurate to at least 3 decimal places.)
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2. Calculate the variance of Z1.
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3. Calculate the covariance of Z1 and Z2.
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It's a simple question.

Think in lines of binomial random variables and basic cov formulas

To calculate the covariance of Y1 and Y2, we need to determine the covariance between X1-X3 and X2-X3.

1. Calculate the covariance of Y1 and Y2:

Cov(Y1, Y2) = Cov(X1-X3, X2-X3)

Since X1, X2, and X3 are independent, we can use the property Cov(A+B, C+D) = Cov(A, C) + Cov(A, D) + Cov(B, C) + Cov(B, D).

Cov(Y1, Y2) = Cov(X1, X2) - Cov(X1, X3) - Cov(X2, X3) + Cov(X3, X3)

Cov(X1, X2) = 0 since they are independent
Cov(X1, X3) = 0 since they are independent
Cov(X2, X3) = 0 since they are independent
Cov(X3, X3) = Var(X3) = np(1-p) = 2*(1/2)*(1-1/2) = 1/2

Thus, Cov(Y1, Y2) = 0 - 0 - 0 + 1/2 = 1/2.

The covariance of Y1 and Y2 is 1/2.

2. Calculate the variance of Z1:

To calculate the variance of Z1, we need to determine the probability P(Z1=1) when Yi=0 for i=1.

Z1=1 if and only if Y1=0. Since Y1 is the difference of two binomial random variables, we can determine the probability P(Y1 = 0).

The probability mass function of a binomial random variable is given by P(X=k) = (n choose k) * p^k * (1-p)^(n-k).

P(Y1 = 0) = P(X1-X3 = 0)
= P(X1 = X3)
= P(X1 = 0, X2 = 0) + P(X1 = 1, X2 = 1)
= (1/2)^2 + (1/2)^2
= 1/4 + 1/4
= 1/2

Since Z1=1 if Y1=0, we have P(Z1=1) = P(Y1=0) = 1/2.

The variance of Z1 is given by Var(Z1) = P(Z1=1)*(1-P(Z1=1)).

Var(Z1) = (1/2)*(1-1/2) = 1/4.

The variance of Z1 is 1/4.

3. Calculate the covariance of Z1 and Z2:

To calculate the covariance of Z1 and Z2, we need to determine the probability P(Z1=1, Z2=1).

Z1=1 if and only if Y1=0, and Z2=1 if and only if Y2=0. Since Y1 and Y2 are independent, we can determine the probability P(Y1 = 0, Y2 = 0) by multiplying the probabilities P(Y1 = 0) and P(Y2 = 0).

P(Y1 = 0, Y2 = 0) = P(Y1 = 0) * P(Y2 = 0)
= (1/2) * (1/2)
= 1/4

Since Z1=1 if Y1=0 and Z2=1 if Y2=0, we have P(Z1=1, Z2=1) = P(Y1=0, Y2=0) = 1/4.

The covariance of Z1 and Z2 is given by Cov(Z1, Z2) = P(Z1=1, Z2=1) - P(Z1=1)*P(Z2=1).

Cov(Z1, Z2) = (1/4) - (1/2) * (1/2)
= 1/4 - 1/4
= 0.

The covariance of Z1 and Z2 is 0.

To calculate the covariance of Y1 and Y2, we need to know the individual variances of Y1 and Y2, as well as the correlation between them.

First, let's calculate the variance of Y1:

Y1 = X1 - X3

Var(Y1) = Var(X1) + Var(X3) - 2 * Cov(X1, X3)

Since X1, X2, and X3 are i.i.d. (independent and identically distributed) Binomial random variables with parameters n=2 and p=1/2:

Var(X1) = Var(X2) = Var(X3) = n * p * (1 - p) = 2 * (1/2) * (1 - 1/2) = 1/2

Cov(X1, X3) = Cov(X2, X3) = 0, since X1, X2, and X3 are independent.

Therefore, Var(Y1) = 1/2 + 1/2 - 2 * 0 = 1

Similarly, let's calculate the variance of Y2:

Y2 = X2 - X3

Var(Y2) = Var(X2) + Var(X3) - 2 * Cov(X2, X3)

Using the same reasoning as above, Var(Y2) = 1

Now, let's calculate the covariance between Y1 and Y2:

Cov(Y1, Y2) = Cov(X1 - X3, X2 - X3)

Cov(Y1, Y2) = Cov(X1, X2) - Cov(X1, X3) - Cov(X2, X3) + Cov(X3, X3)

Since X1, X2, and X3 are independent, Cov(X1, X2) = Cov(X1, X3) = Cov(X2, X3) = 0

Cov(Y1, Y2) = Cov(X3, X3)

Var(X3) = 1/2

Cov(X3, X3) = Var(X3) = 1/2

Therefore, Cov(Y1, Y2) = 1/2

To summarize:

1. The covariance of Y1 and Y2 is 1/2.

Now, let's move on to the other questions.

2. To calculate the variance of Z1, we need to find the probability that Y1 = 0:

Y1 = X1 - X3

When Y1 = 0, X1 = X3

Since X1 and X3 are independent and identically distributed Binomial random variables with parameters n=2 and p=1/2:

P(Y1 = 0) = P(X1 = X3) = P(X1 = 0, X3 = 0) + P(X1 = 1, X3 = 1)

P(X1 = 0, X3 = 0) = P(X1 = 0) * P(X3 = 0) = (1/2)^2 = 1/4

P(X1 = 1, X3 = 1) = P(X1 = 1) * P(X3 = 1) = (1/2)^2 = 1/4

P(Y1 = 0) = 1/4 + 1/4 = 1/2

Since Z1 = 1 if and only if Y1 = 0, the variance of Z1 is equal to the probability of Z1 being 1 multiplied by the probability of Z1 being 0:

Var(Z1) = P(Z1 = 1) * P(Z1 = 0)

P(Z1 = 1) = P(Y1 = 0) = 1/2

P(Z1 = 0) = 1 - P(Z1 = 1) = 1 - 1/2 = 1/2

Var(Z1) = (1/2) * (1/2) = 1/4

3. To calculate the covariance of Z1 and Z2, we need to find the probability that both Y1 and Y2 are equal to 0:

P(Y1 = 0 and Y2 = 0) = P(X1 = X3 and X2 = X3) = P(X1 = 0, X3 = 0, X2 = 0) + P(X1 = 1, X3 = 1, X2 = 1)

P(X1 = 0, X3 = 0, X2 = 0) = P(X1 = 0) * P(X3 = 0) * P(X2 = 0) = (1/2)^3 = 1/8

P(X1 = 1, X3 = 1, X2 = 1) = P(X1 = 1) * P(X3 = 1) * P(X2 = 1) = (1/2)^3 = 1/8

P(Y1 = 0 and Y2 = 0) = 1/8 + 1/8 = 1/4

Since Z1 = 1 if and only if Y1 = 0 and Z2 = 1 if and only if Y2 = 0, the covariance of Z1 and Z2 is equal to the probability of both Z1 and Z2 being 1 minus the product of the probabilities of Z1 being 1 and Z2 being 1:

Cov(Z1, Z2) = P(Z1 = 1 and Z2 = 1) - P(Z1 = 1) * P(Z2 = 1)

P(Z1 = 1 and Z2 = 1) = P(Y1 = 0 and Y2 = 0) = 1/4

P(Z1 = 1) = P(Y1 = 0) = 1/2

P(Z2 = 1) = P(Y2 = 0) = 1/2

Cov(Z1, Z2) = 1/4 - (1/2) * (1/2) = 1/4 - 1/4 = 0

To summarize:

2. The variance of Z1 is 1/4.
3. The covariance of Z1 and Z2 is 0.