4. (10 points) An unfair four-sided dice is thrown 4 times. The probability of getting 1s...

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4. (10 points) An unfair four-sided dice is thrown 4 times. The probability of getting 1s is twice the probability of 2s and half the probability of getting 3s, finally the probability of getting 4s is the same of getting 2s. (a) (3 points) Find the probability that every face appear once. (b) (4 points) Let X be the number of appearances of 2s and Y be the number of appearances of 3s, Z be the number of appearances of the rest. Find the joint distribution of (X,Y) (c) (3 points) Find Cov(X,Y) 4. (10 points) An unfair four-sided dice is thrown 4 times. The probability of getting 1s is twice the probability of 2s and half the probability of getting 3s, finally the probability of getting 4s is the same of getting 2s. (a) (3 points) Find the probability that every face appear once. (b) (4 points) Let X be the number of appearances of 2s and Y be the number of appearances of 3s, Z be the number of appearances of the rest. Find the joint distribution of (X,Y) (c) (3 points) Find Cov(X,Y)