Show that if B is a Brownian motion, then (1 − t)B(t/(1 − t)) and tB(1 − t)/t) are Brownian bridges. In addition, show that if X is a Brownian bridge, then (1+t)X(t/(1+t)) and (1+t)X(1/(1+t)) are...

Show that if B is a Brownian motion, then (1 − t)B(t/(1 − t)) and tB(1 − t)/t) are Brownian bridges. In addition, show that if X is a Brownian bridge, then (1+t)X(t/(1+t)) and (1+t)X(1/(1+t)) are Brownian motions. Hint: Take advantage of the Gaussian property