7 (a) A die is rolled three times with outcomes X1, X2, and X3. Let Y3 be the maximum of the values obtained. Show that \[P (Y3 ≤ j) = p(X1 ≤ j)3. \] Use this to find the distribution of Y3. Does Y3 have a bell-shaped distribution?

  1. Now let Yn be the maximum value when n dice are rolled. Find the distribution of Yn. Is this distribution bell-shaped for large values of n?

Ans.

\[7a. P(Y) \le j = P(X1 \le j, X2 \le j, X3 \le j) = P(X1 \le J)^3. \]

7b. In general, \[ P(Yn \le j) = P(X1 \le j)^3 = \big(\tfrac{j}{n}\big)^n \]

Therefore, P(Yn=j)= j

\[P(Yn = j) = \big(\tfrac{j}{n}\big)^n - \big(\tfrac{j-1}{n}\big)^n \]