I have a question on conditional expectation (I guess):
A message being sent over a communications network is assigned by a router to one of three paths (path 1, path 2, path 3). The nature of the network is such that
- 50% of all messages are routed to path 1,
- 30% are routed to path 2,
- and 20% are routed to path 3.
If routed to
- path 1, then the message has a 75% chance of reaching its destination immediately. Otherwise, the message experiences a $5$-second delay and returns to the router.
- path 2, then the message has a 60% chance of reaching its destination immediately. Otherwise, the message experiences a $10$-second delay and returns to the router.
- path 3, then the message has a 40% chance of reaching its destination immediately. Otherwise, the message experiences a $20$-second delay and returns to the router.
Note that the router cannot distinguish between new messages and messages that have returned from an unsuccessful attempt. Let $X$ denote the time until the message reaches its destination.
(a) Compute the expected value of X .
fot this part I guess that:
- P(Y= path 1) $= 0.5$,
- P(Y= path2 ) $= 0.3$ ,
- P(Y=path 3) $= 0.2$ ,
- E(X|Y=1)$ = 0.25 \times 5$ ,
- E(X|Y=2)$ = 0.4 \times 10$ ,
- E(X|Y=3)$ = 0.6 \times 20$
so $E[X]= \sum_y \{ P(Y =y) \times E[X|Y=y] \}$
(b) Compute the standard deviation of X .
However for this problem, I think I need to get $E[X^2|Y=y]$ but I can't figure a way to do that. And is my modeling for part (a) correct?
Can someone help me with that?