We roll a fair die repeatedly. Let N be the number of rolls needed to see...

Question

Question

Answer 1

Answer #1

N: Number of rolls of the die needed to see the first six.

So N follows a geometric distribution with parameter 1/6: N ~ Geometric (1/6)

E[N] = 1 / (1/6) = 6

Y: Number of 5's in the N-1 rolls prior to the first 6.

Y follows a binomial distribution, with p = 1/5 and q = 4/5

E[Y/N=n] = (n-1)*p = (n-1)/5

So we have E(Y) = E_N [E[Y/N=n] ] = E_N [(n-1)/5 ] = (1/5) x E_N [(n-1)] = (1/5) x [E_N -1] = 1/5) x [6-1] = 5

Cov(Y, N) = E(YN ) − µYµN

= E(Y) E(N) − µYµN

= (5) (6) - (5) (6)

= 0.

Answer 2

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