1.

Let fx( ) and fr( ) be two continuous densities (or two discrete densities) that cross exactly once, in the sense that for a certain c, we have f _x ( x) ? f _y( x) if x < c, and f _x( x) ? f _y( x) if x > c. Show that X ? _st Y. Why do the densities fx ( ) and f _y( ) cross at least once?

2.

Show that if X ~ gamma( ? ?) and Y ~ gamma( ? ?')with ? > ?', then X ? _st Y. The same if Y~ gamma( ?' ?) with ? < ? '.

3.

Prove that the binomial ( n, p) distributions increase in p with respect to stochastic order, by constructing a pair ( X, Y) just as in Example 10.2.2 with X ~ binomial( n, p ₁) and Y ~ binomial( n,p ₂) for p ₁ < p ₂, with additionally Pr[ X ? Y] = 1.

4.

Prove the assertion in the previous exercise with the help of Exercise 10.2.1.

5.

As Exercise 10.2.3, but now for the case that X ~ binomial( n ₁, p) and Y ~ binomial( n ₂. p)...