central limit theorem - dartmouth college
TRANSCRIPT
Central Limit Theorem
§ number of measurements needed to estimate the mean
§ convergence of the estimated mean § a better approximation than the
Chebyshev Inequality (LLN).
Xingru [email protected]
XC 2020
Central Limit Theorem
𝑋! 𝑋"
𝑋# 𝑋$
𝑋% ⋯
𝑋& 𝑋'normal
density function
+
+
+
+
+
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Central Limit Theorem
𝑋! 𝑋"
𝑋# 𝑋$
𝑋% ⋯
𝑋& 𝑋'normal
density function
++
+
+
+
If 𝑆' is the sum of 𝑛 mutually independent random variables, then the distribution function of 𝑆' is well-approximated by a normal density function.
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CENTRAL LIMIT THEOREM FOR BERNOULLI TRIALS
binomial distribution
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Bernoulli Trials
A Bernoulli trials process is a sequence of 𝑛 chance experiments such that
§ Each experiment has two possible outcomes, which we may call success and failure.
§ The probability 𝑝 of success on each experiment is the same for each experiment, and this probability is not affected by any knowledge of previous outcomes. The probability 𝑞 of failure is given by 𝑞 = 1 − 𝑝.
50%Toss a Coinhead & tail
60%Weather Forecast
not rain & rain
12.5%Wheel of Fortune50 points & others
𝒑Bernoulli Trialsoutcome 1 & 2
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Binomial Distribution
§ Let 𝑛 be a positive integer and let 𝑝 be a real number between 0 and 1.
§ Let 𝐵 be the random variable which counts the number of successes in a Bernoulli trials process with parameters 𝑛 and 𝑝.
§ Then the distribution 𝑏(𝑛, 𝑝, 𝑘) of 𝐵 is called the binomial distribution.
Binomial Distribution
𝑏 𝑛, 𝑝, 𝑘 =𝑛𝑘𝑝(𝑞')(
1 2 3 4 5
✗ ✓ ✗ ✗ ✓
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Binomial Distribution
§ Consider a Bernoulli trials process with probability 𝑝 for success on each trial.
§ Let 𝑋* = 1 or 0 according as the 𝑖th outcome is a success or failure.
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋'. Then 𝑆' is the number of successes in 𝑛 trials.
§ We know that 𝑆' has as its distribution the binomial probabilities 𝑏(𝑛, 𝑝, 𝑘).
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𝑛 increases
drifts off to the right
flatten out
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The Standardized Sum of 𝑆4
§ Consider a Bernoulli trials process with probability 𝑝 for success on each trial.
§ Let 𝑋* = 1 or 0 according as the 𝑖th outcome is a success or failure.
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋'. Then 𝑆' is the number of successes in 𝑛 trials.
§ We know that 𝑆' has as its distribution the binomial probabilities 𝑏(𝑛, 𝑝, 𝑘).
§ The standardized sum of 𝑆' is given by
𝑆'∗ =,!)'-'-.
.
𝜇 = ⋯ 𝜎% = ⋯
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The Standardized Sum of 𝑆4
§ Consider a Bernoulli trials process with probability 𝑝 for success on each trial.
§ Let 𝑋* = 1 or 0 according as the 𝑖th outcome is a success or failure.
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋'. Then 𝑆' is the number of successes in 𝑛 trials.
§ We know that 𝑆' has as its distribution the binomial probabilities 𝑏(𝑛, 𝑝, 𝑘).
§ The standardized sum of 𝑆' is given by
𝑆'∗ =,!)'-'-.
.
§ 𝑆'∗ always has expected value 0 and variance 1.𝜇 = 0
𝜎% = 1
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The Standardized Sum of 𝑆4
=𝑆'∗ =𝑆' − 𝑛𝑝𝑛𝑝𝑞
𝑏(𝑛, 𝑝, 1)𝑏(𝑛, 𝑝, 2)
𝑏(𝑛, 𝑝, 3)𝑏(𝑛, 𝑝, 4)
𝑏(𝑛, 𝑝, 5)
𝑏(𝑛, 𝑝, 0)
⋯𝑏(𝑛, 𝑝, 𝑛)
𝑏(𝑛, 𝑝, 𝑘)
0 1 2 3 4 5 … 𝑛
0 − 𝑛𝑝𝑛𝑝𝑞
1 − 𝑛𝑝𝑛𝑝𝑞
2 − 𝑛𝑝𝑛𝑝𝑞
3 − 𝑛𝑝𝑛𝑝𝑞
4 − 𝑛𝑝𝑛𝑝𝑞
5 − 𝑛𝑝𝑛𝑝𝑞
… 𝑛 − 𝑛𝑝𝑛𝑝𝑞
𝑘
𝑥XC 2020
𝑆'∗ =𝑆' − 𝑛𝑝𝑛𝑝𝑞
𝜇 = 0
𝜎% = 1
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standardized sum
standard normal distribution
shapes: sameheights: different
why?
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standardized sum standard normal distribution
shapes: sameheights: different why?
standardized sum: ∑(/0' ℎ* = 1
standard normal distribution: ∫)1
21𝑓 𝑥 𝑑𝑥 = 1
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standardized sum: ∑(/0' ℎ* = 1
standard normal distribution: ∫)1
21𝑓 𝑥 𝑑𝑥 = 1 ≈ ∑(/0' 𝑓 𝑥* 𝑑𝑥
C(/0
'
ℎ* = 1 C(/0
'
𝑓 𝑥* 𝑑𝑥 ≈ 1
=ℎ( = 𝑓 𝑥( 𝑑𝑥
=𝑓 𝑥( =ℎ(𝑑𝑥
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=ℎ) = 𝑏(𝑛, 𝑝, 𝑘)=𝑆'∗ =𝑆' − 𝑛𝑝𝑛𝑝𝑞 =𝑑𝑥 = 𝑥)*& − 𝑥) =
1𝑛𝑝𝑞
𝑏(𝑛, 𝑝, 1)𝑏(𝑛, 𝑝, 2)
𝑏(𝑛, 𝑝, 3)𝑏(𝑛, 𝑝, 4)
𝑏(𝑛, 𝑝, 5)
𝑏(𝑛, 𝑝, 0)
⋯𝑏(𝑛, 𝑝, 𝑛)
0 1 2 3 4 5 … 𝑛
0 − 𝑛𝑝𝑛𝑝𝑞
1 − 𝑛𝑝𝑛𝑝𝑞
2 − 𝑛𝑝𝑛𝑝𝑞
3 − 𝑛𝑝𝑛𝑝𝑞
4 − 𝑛𝑝𝑛𝑝𝑞
5 − 𝑛𝑝𝑛𝑝𝑞
… 𝑛 − 𝑛𝑝𝑛𝑝𝑞
=𝑘 = 𝑛𝑝𝑞𝑥) + 𝑛𝑝 𝑎 : the integer nearest to 𝑎
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ℎ( = 𝑏(𝑛, 𝑝, 𝑘)
𝑑𝑥 = 𝑥(2& − 𝑥( =1𝑛𝑝𝑞
𝑘 = 𝑛𝑝𝑞𝑥( + 𝑛𝑝
=𝑓 𝑥( = 3"45= 𝑛𝑝𝑞 𝑏(𝑛, 𝑝, 𝑛𝑝𝑞𝑥( + 𝑛𝑝 )
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Central Limit Theorem for Binomial Distributions
§ For the binomial distribution 𝑏(𝑛, 𝑝, 𝑘) we have
lim'→1
𝑛𝑝𝑞𝑏 𝑛, 𝑝, 𝑛𝑝 + 𝑥 𝑛𝑝𝑞 = 𝜙(𝑥),
where 𝜙(𝑥) is the standard normal density.
§ The proof of this theorem can be carried out using Stirling’s approximation.
Stirling’s FormulaThe sequence 𝑛! is asymptotically equal
to 𝑛'𝑒)' 2𝜋𝑛.
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Approximating Binomial Distributions
=lim'→1
𝑛𝑝𝑞𝑏 𝑛, 𝑝, 𝑛𝑝 + 𝑥 𝑛𝑝𝑞 = 𝜙(𝑥)
?𝑏 𝑛, 𝑝, 𝑘 ≈ ⋯
normalbinomial
binomial normal
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Approximating Binomial Distributions
=lim'→,
𝑛𝑝𝑞𝑏 𝑛, 𝑝, 𝑛𝑝 + 𝑥 𝑛𝑝𝑞 = 𝜙(𝑥) ?𝑏 𝑛, 𝑝, 𝑘 ≈ ⋯
𝑘 = 𝑛𝑝 + 𝑥 𝑛𝑝𝑞
𝑥 =𝑘 − 𝑛𝑝𝑛𝑝𝑞
𝑏 𝑛, 𝑝, 𝑘 ≈𝜙(𝑥)𝑛𝑝𝑞
𝑏(𝑛, 𝑝, 𝑘) ≈1𝑛𝑝𝑞
𝜙(𝑘 − 𝑛𝑝𝑛𝑝𝑞
)
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Example
Toss a coin
Find the probability of exactly 55 heads in 100 tosses of a coin.
=lim'→,
𝑛𝑝𝑞𝑏 𝑛, 𝑝, 𝑛𝑝 + 𝑥 𝑛𝑝𝑞 = 𝜙(𝑥)
𝑏(𝑛, 𝑝, 𝑘) ≈1𝑛𝑝𝑞
𝜙(𝑘 − 𝑛𝑝𝑛𝑝𝑞
)
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Example
Toss a coin
Find the probability of exactly 55 heads in 100 tosses of a coin.
𝑏(𝑛, 𝑝, 𝑘) ≈1𝑛𝑝𝑞
𝜙(𝑘 − 𝑛𝑝𝑛𝑝𝑞
)𝑛 = 100 𝑝 =12
𝑛𝑝 = 50 𝑛𝑝𝑞 = 5
𝑘 = 55
𝑥 =𝑘 − 𝑛𝑝𝑛𝑝𝑞 = 1 𝑏(100,
12, 55) ≈
15𝜙(1)
XC 2020
Example
Toss a coin
Find the probability of exactly 55 heads in 100 tosses of a coin.
𝑏(𝑛, 𝑝, 𝑘) ≈1𝑛𝑝𝑞 𝜙(
𝑘 − 𝑛𝑝𝑛𝑝𝑞 )
𝑏 100,12, 55 ≈
15𝜙 1 =
15(12𝜋
𝑒)&/%)
Standard normal distribution 𝒁
𝜇 = 0 and 𝜎 = 1
𝑓8 𝑥 = &%9:
𝑒) 5); #/%:# = &%9𝑒)5#/%.
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𝑏(𝑛, 𝑝, 1)𝑏(𝑛, 𝑝, 2)
𝑏(𝑛, 𝑝, 3)𝑏(𝑛, 𝑝, 4)
𝑏(𝑛, 𝑝, 5)
𝑏(𝑛, 𝑝, 0)
⋯𝑏(𝑛, 𝑝, 𝑛)
0 1 2 3 4 5 … 𝑛
0 − 𝑛𝑝𝑛𝑝𝑞
1 − 𝑛𝑝𝑛𝑝𝑞
2 − 𝑛𝑝𝑛𝑝𝑞
3 − 𝑛𝑝𝑛𝑝𝑞
4 − 𝑛𝑝𝑛𝑝𝑞
5 − 𝑛𝑝𝑛𝑝𝑞
… 𝑛 − 𝑛𝑝𝑛𝑝𝑞
A specific outcome
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𝑏(𝑛, 𝑝, 1)𝑏(𝑛, 𝑝, 2)
𝑏(𝑛, 𝑝, 3)𝑏(𝑛, 𝑝, 4)
𝑏(𝑛, 𝑝, 5)
𝑏(𝑛, 𝑝, 0)
⋯𝑏(𝑛, 𝑝, 𝑛)
0 1 2 3 4 5 … 𝑛
0 − 𝑛𝑝𝑛𝑝𝑞
1 − 𝑛𝑝𝑛𝑝𝑞
2 − 𝑛𝑝𝑛𝑝𝑞
3 − 𝑛𝑝𝑛𝑝𝑞
4 − 𝑛𝑝𝑛𝑝𝑞
5 − 𝑛𝑝𝑛𝑝𝑞
… 𝑛 − 𝑛𝑝𝑛𝑝𝑞
Outcomes in an Interval
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Central Limit Theorem for Binomial Distributions
§ Let 𝑆' be the number of successes in n Bernoulli trials with probability 𝑝 for success, and let 𝑎 and 𝑏 be two fixed real numbers.
§ Then
lim'→21
𝑃 𝑎 ≤ ,!)'-'-.
≤ 𝑏 = ∫<=𝜙 𝑥 𝑑𝑥 = NA(𝑎, 𝑏).
§ We denote this area by NA(𝑎, 𝑏).
𝑎 ≤𝑆' − 𝑛𝑝𝑛𝑝𝑞
= 𝑆'∗ ≤ 𝑏 𝑎 𝑛𝑝𝑞 + 𝑛𝑝 ≤ 𝑆' ≤ 𝑏 𝑛𝑝𝑞 + 𝑛𝑝
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lim'→21
𝑃 𝑎 ≤ ,!)'-'-.
≤ 𝑏 = NA(𝑎, 𝑏).
𝑎 ≤𝑆' − 𝑛𝑝𝑛𝑝𝑞
= 𝑆'∗ ≤ 𝑏 𝑎 𝑛𝑝𝑞 + 𝑛𝑝 ≤ 𝑆' ≤ 𝑏 𝑛𝑝𝑞 + 𝑛𝑝
𝑖 ≤ 𝑆' ≤ 𝑗 𝑎 =𝑖 − 𝑛𝑝𝑛𝑝𝑞
≤ 𝑆'∗ ≤𝑗 − 𝑛𝑝𝑛𝑝𝑞
= 𝑏
lim'→21
𝑃 𝑖 ≤ 𝑆' ≤ 𝑗 = lim'→21
𝑃 *)'-'-.
≤ 𝑆'∗ ≤>)'-'-.
= NA(*)'-'-.
, >)'-'-.
).
Approximating Binomial Distributions
XC 2020
𝑖 ≤ 𝑆' ≤ 𝑗 𝑎 =𝑖 − 𝑛𝑝𝑛𝑝𝑞
≤ 𝑆'∗ ≤𝑗 − 𝑛𝑝𝑛𝑝𝑞
= 𝑏
lim'→21
𝑃 𝑖 ≤ 𝑆' ≤ 𝑗 = lim'→21
𝑃 *)'-'-.
≤ 𝑆'∗ ≤>)'-'-.
= NA(*)'-'-.
, >)'-'-.
).
Approximating Binomial Distributions (more accurate)
lim@→BC
𝑃 𝑖 ≤ 𝑆@ ≤ 𝑗 = NA(DE!"E@F
@FG,HB!"E@F
@FG).
𝑖 𝑗
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Example
Toss a coin
A coin is tossed 100 times. Estimate the probability that the number of heads lies between 40 and 60 (including the end points).
=lim'→21
𝑃 𝑖 ≤ 𝑆' ≤ 𝑗 = NA(𝑖 − 12 − 𝑛𝑝
𝑛𝑝𝑞 ,𝑗 + 12 − 𝑛𝑝
𝑛𝑝𝑞 )
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Toss a coin
A coin is tossed 100 times. Estimate the probability that the number of heads lies between 40 and 60 (including the end points).
𝑛 = 100 𝑝 =12
𝑛𝑝 = 50 𝑛𝑝𝑞 = 5
𝑖 = 40
𝑖 − 12 − 𝑛𝑝𝑛𝑝𝑞
= −2.1
𝑗 = 60
𝑗 + 12 − 𝑛𝑝𝑛𝑝𝑞
= 2.1
the outcome will not deviate by more than two standard deviations from the expected value
XC 2020
Toss a coin
A coin is tossed 100 times. Estimate the probability that the number of heads lies between 40 and 60 (including the end points).
𝑖 = 40
𝑖 − 12 − 𝑛𝑝𝑛𝑝𝑞 = −2.1
𝑗 = 60
𝑗 + 12 − 𝑛𝑝𝑛𝑝𝑞 = 2.1
=lim$→&'
𝑃 𝑖 ≤ 𝑆$ ≤ 𝑗 = NA(𝑖 − 12 − 𝑛𝑝
𝑛𝑝𝑞,𝑗 + 12 − 𝑛𝑝
𝑛𝑝𝑞)
=lim$→&'
𝑃 40 ≤ 𝑆$ ≤ 60 = NA −2.1,2.1= 2NA 0, 2.1
2𝜎 2.1𝜎
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Approximating Binomial Distribution
𝑺𝒏∗ = 𝒙lim$→'
𝑛𝑝𝑞𝑏 𝑛, 𝑝, 𝑛𝑝 + 𝑥 𝑛𝑝𝑞 = 𝜙(𝑥)
𝑺𝒏 = 𝒌
𝑏(𝑛, 𝑝, 𝑘) ≈1𝑛𝑝𝑞 𝜙(
𝑘 − 𝑛𝑝𝑛𝑝𝑞 )
𝒂 ≤ 𝑺𝒏∗ ≤ 𝒃
lim'→*,
𝑃 𝑎 ≤ -!.'/'/0
≤ 𝑏 = NA(𝑎, 𝑏).
𝒊 ≤ 𝑺𝒏 ≤ 𝒋
lim$→&'
𝑃 𝑖 ≤ 𝑆$ ≤ 𝑗 = NA(()!")$*
$*+,,&!")$*
$*+).
CLT
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CENTRAL LIMIT THEOREM FOR DISCRETE INDEPENDENT TRIALS
any independent trials process such that the individual trials have finite variance
XC 2020
The Standardized Sum of 𝑆4
§ Consider an independent trials process with common distribution function 𝑚 𝑥defined on the integers, with expected value 𝜇 and variance 𝜎%.
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋' be the sum of 𝑛 independent discrete random variables of the process.
§ The standardized sum of 𝑆' is given by
𝑆'∗ =,!)';':#
.
§ 𝑆'∗ always has expected value 0 and variance 1.
𝐸(𝑆') = ⋯
𝑉(𝑆') = ⋯
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independent trials process
𝑃(𝑆' = 𝑘)
𝑘 𝑥)=𝑘 − 𝑛𝜇𝑛𝜎%
=𝑆'∗ =𝑆' − 𝑛𝜇𝑛𝜎%
normal
𝑛𝜎-𝑃(𝑆$ = 𝑘)
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Approximation Theorem
§ Let 𝑋&, 𝑋%, ⋯, 𝑋' be an independent trials process and let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋'.
§ Assume that the greatest common divisor of the differences of all the values that the 𝑋( can take on is 1.
§ Let 𝐸 𝑋( = 𝜇 and 𝑉 𝑋( = 𝜎%.
§ Then for 𝑛 large,
𝑃(𝑆' = 𝑘) ≈ &':#
𝜙(()';':#
).
§ Here 𝜙(𝑥) is the standard normal density.
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Central Limit Theorem for a Discrete Independent Trials Process
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋' be the sum of n discrete independent random variables with common distribution having expected value 𝜇 and variance 𝜎%.
§ Then, for 𝑎 < 𝑏,
lim'→21
𝑃 𝑎 ≤ ,!)';':#
≤ 𝑏 = &%9∫<
= 𝑒)5#/%𝑑𝑥 = NA(𝑎, 𝑏).
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Approximating a Discrete Independent Trials Process
𝑃(𝑆' = 𝑘) ≈1𝑛𝜎%
𝜙(𝑘 − 𝑛𝜇𝑛𝜎%
)
lim'→*,
𝑃 𝑐 ≤ 𝑆' ≤ 𝑑 =
NA(1.'2'3"
, 4.'2'3"
).
𝑺𝒏 = 𝒌
𝒄 ≤ 𝑺𝒏 ≤ 𝒅
CLT
XC 2020
Approximating a Discrete Independent Trials Process
lim'→*,
𝑃 𝑖 ≤ 𝑆' ≤ 𝑗 =
NA(5.#".'/
'/0,6*#".'/
'/0).
𝒊 ≤ 𝑺𝒏 ≤ 𝒋
𝑃(𝑆' = 𝑘) ≈1𝑛𝜎%
𝜙(𝑘 − 𝑛𝜇𝑛𝜎%
)
𝑺𝒏 = 𝒌
𝑏(𝑛, 𝑝, 𝑘) ≈1𝑛𝑝𝑞
𝜙(𝑘 − 𝑛𝑝𝑛𝑝𝑞
)
𝑺𝒏 = 𝒌
lim'→*,
𝑃 𝑐 ≤ 𝑆' ≤ 𝑑 =
NA(1.'2'3"
, 4.'2'3"
).
𝒄 ≤ 𝑺𝒏 ≤ 𝒅
Bernoulli
XC 2020
ExampleA surveying instrument makes an error of -2, -1, 0, 1, or 2 feet with equal probabilities when measuring the height of a 200-foot tower.
XC 2020
Find the expected value and the variance for the height obtained using this instrument once.
Example
height = error + 200
error -2 -1 0 1 2
probability15
15
15
15
15
𝐸 error =15−2 − 1 + 0 + 1 + 2 = 0
𝑉 error =15(−2)%+(−1)%+0% + 1% + 2% = 2
𝐸 height= 0 + 200 = 200
𝑉 height = 2
XC 2020
Estimate the probability that in 18 independent measurements of this tower, the average of the measurements is between 199 and 201, inclusive.
Example
𝜇 = 200 𝜎% = 2
=lim'→21
𝑃 𝑐 ≤ 𝑆' ≤ 𝑑 = NA(𝑐 − 𝑛𝜇𝑛𝜎%
,𝑑 − 𝑛𝜇𝑛𝜎%
).
199 ≤𝑆'𝑛=𝑆'18
≤ 201𝑛 = 18
XC 2020
Example
𝜇 = 200 𝜎% = 2
=lim'→*,
𝑃 𝑐 ≤ 𝑆' ≤ 𝑑 = NA(𝑐 − 𝑛𝜇𝑛𝜎%
,𝑑 − 𝑛𝜇𝑛𝜎%
).
199 ≤𝑆'𝑛=𝑆'18
≤ 201𝑛 = 18
𝑃 199 ≤𝑆'18
≤ 201 = 𝑃 3582 ≤ 𝑆' ≤ 3618
≈ NA3582 − 3600
36,3618 − 3600
36= NA(−3, 3).
XC 2020
03
01
02
§ independent
Discrete Random Variables
§ independent§ identical
Bernoulli Trials
§ independent§ identical
Discrete Trials
Central Limit Theorem
XC 2020
Central Limit Theorem
§ Let 𝑋&, 𝑋%, ⋯, 𝑋' be a sequence of independent discrete random variables. There exist a constant 𝐴, such that 𝑋* ≤ 𝐴 for all 𝑖.
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋' be the sum. Assume that 𝑆' → ∞.
§ For each 𝑖, denote the expected value and variance of 𝑋* by 𝜇* and 𝜎*%, respectively.
§ Define the expected value and variance of 𝑆' to be 𝑚' and 𝑠'%, respectively.
§ For 𝑎 < 𝑏,
lim'→21
𝑃 𝑎 ≤ ,!)@!A!
≤ 𝑏 = &%9∫<
= 𝑒)5#/%𝑑𝑥 = NA(𝑎, 𝑏).
XC 2020
CENTRAL LIMIT THEOREM FOR CONTINUOUS INDEPENDENT TRIALScontinuous random variables with a common density function
XC 2020
Central Limit Theorem
§ Let 𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋' be the sum of 𝑛 independent continuous random variables with common density function 𝑝 having expected value 𝜇 and variance 𝜎%.
§ Let
𝑆'∗ =,!)';':#
.
§ Then we have, for all 𝑎 < 𝑏,
lim'→21
𝑃 𝑎 ≤ 𝑆'∗ ≤ 𝑏 = &%9∫<
= 𝑒)5#/%𝑑𝑥 = NA(𝑎, 𝑏).
XC 2020
Example
§ Suppose a surveyor wants to measure a known distance, say of 1 mile, using a transit and some method of triangulation.
§ He knows that because of possible motion of the transit, atmospheric distortions, and human error, any one measurement is apt to slightly in error.
§ He plans to make several measurements and take an average.
§ He assumes that his measurements are independent random variables with common distribution of mean 𝜇 = 1 and standard deviation 𝜎 =0.0002.
XC 2020
Example
§ He can say that if 𝑛 is large, the average ,!
'has a density function that
is approximately normal, with mean 1mile, and standard deviation 0.000%
'miles.
§ How many measurements should he make to be reasonably sure that his average lies within 0.0001 of the true value?
XC 2020
Example
𝜇 = 1
Expected Value
𝜎 = 0.0002
Standard Deviation𝑆' = 𝑋& + 𝑋% +⋯+ 𝑋'
Sum of 𝒏 independent measurements
§ He can say that if 𝑛 is large, the average ,!' has a density function that is approximately normal, with mean 1 mile, and standard deviation 0.000%
'miles.
𝜎U = (0.0002)UVariance 𝐴' =
𝑋& + 𝑋% +⋯+ 𝑋'𝑛
𝐸 𝐴' = 𝜇𝐷 𝐴' =
𝜎𝑛
𝑉 𝐴' =𝜎%
𝑛
Average of 𝒏 independent measurements
XC 2020
Example
Chebyshev inequality
LLN
𝑆'∗ =𝑆' − 𝑛𝜇𝑛𝜎%
CLT
§ How many measurements should he make to be reasonably sure that his average lies within 0.0001 of the true value?
XC 2020
𝝈𝟐
𝜺𝟐
𝑷(|𝑿 − 𝝁|≥ 𝜺)
inequalityChebyshev𝑷(|𝑿 − 𝝁| ≥ 𝜺) ≤
𝝈𝟐
𝜺𝟐
This is a sample text. Insert your desired text
here.
This is a sample text. Insert your desired text here.
Let 𝑋 be a discrete random variable with expected value 𝜇 = 𝐸(𝑋) and variance 𝜎% = 𝑉(𝑋), and let 𝜀 > 0 be any positive number. Then
not necessarily positive
XC 2020
Example
Chebyshev inequality
LLN
§ How many measurements should he make to be reasonably sure that his average lies within 0.0001 of the true value?
𝜇 = 1 𝜎% = (0.0002)%
=𝑃(|𝑋 − 𝜇| ≥ 𝜀) ≤𝜎%
𝜀%
𝑃 𝐴' − 𝜇 ≥ 0.0001 ≤𝜎%
𝑛𝜀%=
0.0002 %
𝑛 0.0001 % =4𝑛
XC 2020
Example
Chebyshev inequality
LLN
§ How many measurements should he make to be reasonably sure that his average lies within 0.0001 of the true value?
=𝑃(|𝑋 − 𝜇| ≥ 𝜀) ≤𝜎%
𝜀%
𝑃 𝐴' − 𝜇 ≥ 0.0001 ≤𝜎%
𝑛𝜀%=4𝑛= 0.05 𝑛 = 80
XC 2020
Example
𝑆'∗ =𝑆' − 𝑛𝜇𝑛𝜎%
CLT
=lim'→21
𝑃 𝑎 ≤ 𝑆'∗ ≤ 𝑏 = NA(𝑎, 𝑏)
𝑃 𝐴' − 𝜇 ≤ 0.0001 = 𝑃 𝑆' − 𝑛𝜇 ≤ 0.5𝑛𝜎
= 𝑃𝑆' − 𝑛𝜇𝑛𝜎%
≤ 0.5 𝑛 = 𝑃 −0.5 𝑛 ≤ 𝑆'∗ ≤ 0.5 𝑛
≈ NA −0.5 𝑛, 0.5 𝑛
𝜇 = 1 𝜎% = (0.0002)%
XC 2020
Example
𝑆'∗ =𝑆' − 𝑛𝜇𝑛𝜎%
CLT
=lim'→21
𝑃 𝑎 ≤ 𝑆'∗ ≤ 𝑏 = NA(𝑎, 𝑏)
𝑃 𝐴' − 𝜇 ≤ 0.0001 ≈ NA −0.5 𝑛, 0.5 𝑛 = 0.95
0.5 𝑛 = 2 𝑛 = 16
XC 2020
Example
Chebyshev inequality
LLN
𝑆'∗ =𝑆' − 𝑛𝜇𝑛𝜎%
CLT
§ How many measurements should he make to be reasonably sure that his average lies within 0.0001 of the true value?
𝑛 = 80 𝑛 = 16
XC 2020