Consider n independent rolls of a k-sided fair die with k≥2: the sides of the die are labelled 1,2,…,k and each side has probability 1/k of facing up after a roll. Let the random variable Xi denote the number of rolls that result in side i facing up. Thus, the random vector (X1,…,Xk) has a multinomial distribution.

1. Which of the following statements is correct? Try to answer without doing any calculations.

- unanswered X1 and X2 are uncorrelated.X1 and X2 are positively correlated.X1 and X2 are negatively correlated.
2. Find the covariance, cov(X1,X2), of X1 and X2. Express your answer as a function of n and k using standard notation. Hint: Use indicator variables to encode the result of each roll.

cov(X1,X2)= - unanswered
3. Suppose now that the die is biased, with a probability pi≠0 that the result of any given die roll is i, for i=1,2,…,k. We still consider n independent tosses of this biased die and define Xi to be the number of rolls that result in side i facing up.

Generalize your answer to part 2: Find cov(X1,X2) for this case of a biased die. Express your answer as a function of n,k,p1,p2 using standard notation. Write p1 and p2 as 'p_1' and 'p_2', respectively, and wrap them in parentheses in your answer; i.e., enter '(p_1)' and '(p_2)'.

cov(X1,X2)= - unanswered

3. -n*(p_1)*(p_2)

by the book solution. Instead of p_i or p_j we are given p_1 and p_2. In any case, k is useless

hi there Juan Pro and Anonymous...First and foremost if you are able to provide the answers to the rest of this problem set....you are more than welcome

Thanks for providing the answers to questions 1 and 2

1. x1 x2 are negative correlated

2. covariance is : -n/(k^2)
3. unaswered (if anyone please post)!

3. -n*p_1*p_2

1. X1 and X2 are uncorrelated.

2. cov(X1, X2) = -(n/k^2).

3. cov(X1, X2) = -n(p1)(p2).

1. Without doing any calculations, we can analyze the relationship between X1 and X2 based on their definition. Since each roll of the die is independent, the outcome of one roll does not affect the outcome of another roll. Therefore, X1 and X2 are uncorrelated.

2. To find the covariance cov(X1, X2), we can use the definition of covariance. Covariance measures how two random variables vary together. In this case, we can express X1 and X2 as indicator variables that encode the result of each roll.

Let's use I(i) to represent the indicator variable for the event that side i facing up after a roll. The indicator variable takes the value 1 if the event occurs and 0 otherwise. In our case, X1 and X2 can be defined as follows:

X1 = I(1) + I(1) + ... + I(1) (sum of n independent indicator variables for event 1)
X2 = I(2) + I(2) + ... + I(2) (sum of n independent indicator variables for event 2)

The covariance of X1 and X2 can be calculated as:

cov(X1, X2) = E[X1*X2] - E[X1]*E[X2]

We need to calculate the expected values of X1*X2, E[X1], and E[X2] to compute the covariance.

3. Generalizing the answer to part 2 with a biased die involves considering the probabilities of each side facing up. Let p1 and p2 represent the probabilities that side 1 and side 2 face up after a roll, respectively. For a biased die, we have:

X1 = I(1) + I(1) + ... + I(1) (sum of n independent indicator variables for event 1)
X2 = I(2) + I(2) + ... + I(2) (sum of n independent indicator variables for event 2)

Using similar steps as in part 2, we can calculate the covariance cov(X1, X2) for this case:

cov(X1, X2) = E[X1*X2] - E[X1]*E[X2]

The expected values E[X1*X2], E[X1], and E[X2] can be calculated considering the probabilities pi for each side facing up.

Therefore, to fully answer part 2 and part 3, the specific values of n, k, p1, and p2 need to be provided to calculate the covariance.