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.

Question

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

Answer 1

1. x1 x2 are negative correlated
2. covariance is : -n/(k^2)
3. unaswered (if anyone please post)!

Answer 2

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

Answer 3

3. -n*p_1*p_2