Probability - Random VariablesWatch

Announcements
This discussion is closed.
Thread starter 14 years ago
#1 help

Hmm, probability just got difficult, so I thought I'd see if anyone has any ideas about this:

Let X_1, . . . ,X_N be independent identically distributed random variables, where N is a non-negative integer-valued random variable. Let Z = X_1 + . . . + X_N, (assuming that Z = 0 if N = 0).

a) Find E(Z)

b)Show that V(Z) = V(N)E(X_1)^2 + E(N)V(X_1)

???
0
14 years ago
#2
E(Z)
= (sum over n) E(Z | N = n) P(N = n)
= (sum over n) n E(X_1) P(N = n)
= E(X_1) * (sum over n) n P(N = n)
= E(X_1) E(N)

For the next calculation we need to know that E(X_i X_j) = E(X_i)E(X_j) = E(X_1)^2 for all distinct i and j, which follows from the independence of X_i and X_j.

E(Z^2)
= (sum over n) E(Z^2 | N = n) P(N = n)
= (sum over n) E[(X_1 + ... + X_n)^2] P(N = n)
= (sum over n) [n E(X_1^2) + n(n - 1)E(X_1)^2] P(N = n)
= (sum over n) [n (E(X_1^2) - E(X_1)^2) + n^2 E(X_1)^2] P(N = n)
= (sum over n) [n V(X_1) + n^2 E(X_1)^2] P(N = n)
= V(X_1) * (sum over n) n P(N = n)
= + E(X_1)^2 * (sum over n) n^2 P(N = n)
= V(X_1) E(N) + E(X_1)^2 E(N^2)

V(Z)
= E(Z^2) - E(Z)^2
= V(X_1) E(N) + E(X_1)^2 E(N^2) - E(X_1)^2 E(N)^2
= V(X_1) E(N) + E(X_1)^2 (E(N^2) - E(N)^2)
= V(X_1) E(N) + E(X_1)^2 V(N)
0
X
new posts Back
to top
Latest
My Feed

Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

University open days

• Bournemouth University
Midwifery Open Day at Portsmouth Campus Undergraduate
Wed, 16 Oct '19
• Teesside University
All faculties open Undergraduate
Wed, 16 Oct '19
• University of the Arts London
London College of Fashion – Cordwainers Footwear and Bags & Accessories Undergraduate
Wed, 16 Oct '19

Poll

Join the discussion

How has the start of this academic year been for you?

Loving it - gonna be a great year (110)
17.92%
It's just nice to be back! (166)
27.04%
Not great so far... (221)
35.99%
I want to drop out! (117)
19.06%