題組內容

1. (15%) Let \( X_1, X_2, X_3 \) denote a random sample of size \( n = 3 \) from a distribution with the p.d.f. \[ f(x) = \left(\frac{2}{3}\right)\left(\frac{1}{3}\right)^{x-1}, x=1, 2, 3, \dots . \] That is, \( X_1, X_2, X_3 \) are mutually independent and each has the geometric distribution.

(a) (5%) Compute \( P(X_1=1, X_2=2, X_3=1) \).