LECTURE 19
Limit theorems – I
Chebyshev’s inequality
• Random variable X
(with finite mean µ and variance σ 2)
• Readings: Sections 5.1-5.3;
start Section 5.4
σ2 =
• X1, . . . , Xn i.i.d.
≥
X1 + · · · + Xn
n
What happens as n → ∞?
Mn =
!
(x − µ)2fX (x) dx
! −c
−∞
(x − µ)2fX (x) dx +
! ∞
c
(x − µ)2fX (x) dx
≥ c2 · P(|X − µ| ≥ c)
• Why bother?
σ2
c2
P(|X − µ| ≥ c) ≤
• A tool: Chebyshev’s inequality
• Convergence “in probability”
P(|X − µ| ≥ kσ ) ≤
• Convergence of Mn
(weak law of large numbers)
Deterministic limits
1
k2
Convergence “in probability”
• Sequence an
Number a
• Sequence of random variables Yn
• converges in probability to a number a:
“(almost all) of the PMF/PDF of Yn ,
eventually gets concentrated
(arbitrarily) close to a”
• an converges to a
lim a = a
n→∞ n
“an eventually gets and stays
(arbitrarily) close to a”
• For every " > 0,
lim P(|Yn − a| ≥ ") = 0
n→∞
• For every " > 0,
there exists n0,
such that for every n ≥ n0,
we have |an − a| ≤ ".
1 - 1 /n
pmf of Yn
1 /n
0
Does Yn converge?
1
n
Convergence of the sample mean
(Weak law of large numbers)
The pollster’s problem
• f : fraction of population that “. . . ”
• X1, X2, . . . i.i.d.
finite mean µ and variance σ 2
• ith (randomly selected) person polled:
X + · · · + Xn
Mn = 1
n
Xi =

1,
0,
if yes,
if no.
• Mn = (X1 + · · · + Xn)/n
fraction of “yes” in our sample
• E[Mn] =
• Goal: 95% confidence of ≤1% error
• Var(Mn) =
P(|Mn − f | ≥ .01) ≤ .05
P(|Mn − µ| ≥ ") ≤
• Use Chebyshev’s inequality:
σ2
Var(Mn)
=
"2
n"2
P(|Mn − f | ≥ .01) ≤
2
σM
n
(0.01)2
σx2
1
=
≤
4n(0.01)2
n(0.01)2
• Mn converges in probability to µ
• If n = 50, 000,
then P(|Mn − f | ≥ .01) ≤ .05
(conservative)
Different scalings of Mn
The central limit theorem
• X1, . . . , Xn i.i.d.
finite variance σ 2
• “Standardized” Sn = X1 + · · · + Xn:
Zn =
• Look at three variants of their sum:
• Sn = X1 + · · · + Xn
– zero mean
variance nσ 2
– unit variance
Sn
variance σ 2/n
n
converges “in probability” to E[X] (WLLN)
• Let Z be a standard normal r.v.
(zero mean, unit variance)
• Mn =
Sn
• √
n
Sn − nE[X]
Sn − E[Sn]
=
√
σSn
nσ
• Theorem: For every c:
constant variance σ 2
P(Zn ≤ c) → P(Z ≤ c)
– Asymptotic shape?
• P(Z ≤ c) is the standard normal CDF,
Φ(c), available from the normal tables
2
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6.041 / 6.431 Probabilistic Systems Analysis and Applied Probability
Fall 2010
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