University of California - Berkeley
Department of Electrical Engineering & Computer Sciences
EE126 Probability and Random Processes
(Spring 2012)
Discussion 12 Notes
April 12, 2012
1. Discussion 11 Notes, Problem 2.
2. Let X1 , · · · , X1 0 be independent random variables, uniformly distributed over the unit interval [0, 1].
Let X = X1 + · · · + X10 . Estimate P (X ≥ 7) using:
(a) Markov inequality
(b) Chebychev inequality
(c) central limit theorem
Solutions:
1. See Discussion 11 Notes.
2. (a)
E[X] = 10E[Xi ] = 5
The Markov inequality gives:
P (X ≥ 7) ≤
5
≈ 0.714.
7
(b)
var(X) = 10var(Xi ) = 10 ·
1
5
=
12
6
We then have:
1
P (|X − 5| ≥ 2)
2
1 var(X)
5
·
=
≈ 0.104
2
4
48
P (X ≥ 7) =
=
where the first equality comes from symmetry, and the inequality from Chebyshev.
p
(c) To use the central limit theorem, assume X is roughly normal. Then let Z = (X−E[X])/ var(x),
which is a standard normal. We then have:
P (X ≥ 7) = 1 − P (X < 7)
!
7−5
= 1−P Z < p
5/6
≈ 1 − Φ(2.19) ≈ 0.0143.
1