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Chapter 6: Statistics and Sampling Distributions
MATH450
September 12th, 2017
MATH450 Chapter 6: Statistics and Sampling Distributions
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Where are we?
Week 1 · · · · · ·• Chapter 1: Descriptive statistics
Week 2 · · · · · ·• Chapter 6: Statistics and SamplingDistributions
Week 4 · · · · · ·• Chapter 7: Point Estimation
Week 7 · · · · · ·• Chapter 8: Confidence Intervals
Week 10 · · · · · ·• Chapter 9: Test of Hypothesis
Week 13 · · · · · ·• Two-sample inference, ANOVA, regression
MATH450 Chapter 6: Statistics and Sampling Distributions
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Where are we?
6.1 Statistics and their distributions
6.2 The distribution of the sample mean
6.3 The distribution of a linear combination
Order: 6.1 → 6.3 → 6.2
MATH450 Chapter 6: Statistics and Sampling Distributions
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Random sample
Definition
The random variables X1,X2, ...,Xn are said to form a (simple)random sample of size n if
1 the Xi ’s are independent random variables
2 every Xi has the same probability distribution
MATH450 Chapter 6: Statistics and Sampling Distributions
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Statistics
Definition
A statistic is any quantity whose value can be calculated fromsample data
the probability distribution of a statistic is sometimes referredto as its sampling distribution
MATH450 Chapter 6: Statistics and Sampling Distributions
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Questions for this chapter
Given a random sample X1,X2, . . . ,Xn, and
T = a1X1 + a2X2 + . . .+ anXn
Last week: If we know the distribution of Xi ’s, can we obtainthe distribution of T?
(Today: What if X ′i s follow normal distributions?)
Next Thursday: If we don’t know the distribution of Xi ’s,can we still obtain/approximate the distribution of T?
(Today: Can we at least compute the mean and the variance?)
MATH450 Chapter 6: Statistics and Sampling Distributions
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Moment generating function
MATH450 Chapter 6: Statistics and Sampling Distributions
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Moment generating function
Definition
The moment generating function (mgf) of a continuous randomvariable X is
MX (t) = E (etX ) =
∫ ∞−∞
etx fX (x)dx
Why is it call the moment generating function?
MX (0) =∫∞−∞ fX (x)dx = 1.
M ′X (0) =∫∞−∞ xfX (x)dx = E (X ).
M ′′X (0) =∫∞−∞ x
2fX (x)dx = E (X2).
MATH450 Chapter 6: Statistics and Sampling Distributions
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Moment generating function
Property
Two distributions have the same pdf if and only if they have thesame moment generating function
MATH450 Chapter 6: Statistics and Sampling Distributions
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Moment generating function
MATH450 Chapter 6: Statistics and Sampling Distributions
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Moment generating function
Property
Let X1,X2 be a 2 independent random variables and T = X1 + X2,then
MT (t) = MX1(t)MX2(t)
Hint:
MT (t) = E (etT ) = E (et(X1+X2)) = E (etX1 · etX2)
MATH450 Chapter 6: Statistics and Sampling Distributions
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Example
Problem
Given that the mgf of a normal random variables with mean µ andvariance σ2 is
eµt+σ2
2t2
Suppose X and Y are independent normal random variables. Showthat T = X + Y also follows the normal distribution.
MATH450 Chapter 6: Statistics and Sampling Distributions
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Moment generating function
Property
Let X1,X2, . . . ,Xn be independent random variables with momentgenerating functions MX1(t),MX2(t), . . . ,MXn(t), respectively.Define
T = a1X1 + a2X2 + . . .+ anXn
where a1, a2, . . . , an are constants.Then
MT (t) = MX1(a1t)MX2(a2t) . . .MXn(ant)
Idea:
MT (t) = E (etT ) = E (et(a1X1+a2X2+...+anXn))
= E (et(a1X1) · et(a2X2) · . . . · et(anXn))
MATH450 Chapter 6: Statistics and Sampling Distributions
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Linear combination of normal random variables
Theorem
Let X1,X2, . . . ,Xn be independent normal random variables (withpossibly different means and/or variances). Then
T = a1X1 + a2X2 + . . .+ anXn
also follows the normal distribution.
Hint: the mgf of a normal random variables with mean µ andvariance σ2 is
eµt+σ2
2t2
MATH450 Chapter 6: Statistics and Sampling Distributions
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Linear combination of normal random variables
Theorem
Let X1,X2, . . . ,Xn be independent normal random variables (withpossibly different means and/or variances). Then
T = a1X1 + a2X2 + . . .+ anXn
also follows the normal distribution.
What are the mean and the standard deviation of T?
E (T ) = a1E (X1) + a2E (X2) + . . .+ anE (Xn)
σ2T = a21σ
2X1
+ a22σ2X2
+ . . .+ a2nσ2Xn
MATH450 Chapter 6: Statistics and Sampling Distributions
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Example 1
Problem
Assume that
X1 ∼ N (10, 9) and X2 ∼ N (30, 16)
are independent.
What is the distribution of X1 − X2?
MATH450 Chapter 6: Statistics and Sampling Distributions
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Example 2
Problem
A concert has three pieces of music to be played beforeintermission. The time taken to play each piece has a normaldistribution.Assume that the three times are independent of each other. Themean times are 15, 30, and 20 min, respectively, and the standarddeviations are 1, 2, and 1.5 min, respectively.
What is the distribution of the length of the concert?
MATH450 Chapter 6: Statistics and Sampling Distributions
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How to do computations with normal random variables?
Reading: 4.3
MATH450 Chapter 6: Statistics and Sampling Distributions
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N (µ, σ2)
MATH450 Chapter 6: Statistics and Sampling Distributions
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Basic properties
E (X ) = µ, Var(X ) = σ2
Density function
f (x , µ, σ) =1√
2πσ2e−
(x−µ)2
2σ2
Z = N (0, 1) is called the standard normal distribution
MATH450 Chapter 6: Statistics and Sampling Distributions
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Standard normal distribution
E (Z ) = 0, Var(Z ) = 1
Density function
f (z , 0, 1) =1√2π
e−z2
2
The cumulative distribution function of the standard normaldistribution is:
Φ(z) = P(Z ≤ z) =∫ z−∞
f (y , 0, 1) dy
MATH450 Chapter 6: Statistics and Sampling Distributions
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Φ(z)
Φ(z) = P(Z ≤ z) =∫ z−∞
f (y , 0, 1) dy
MATH450 Chapter 6: Statistics and Sampling Distributions
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Φ(z)
MATH450 Chapter 6: Statistics and Sampling Distributions
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Shifting and scaling normal random variables
Problem
Let X be a normal random variable with mean µ and standarddeviation σ. Then
Z =X − µσ
follows the normal distribution with mean 0 and variance 1.
Hint:MZ (t) = E (e
tZ ) = E (etX−µσ ) = E (e
tσX · et
µσ )
MATH450 Chapter 6: Statistics and Sampling Distributions
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Shifting and scaling normal random variables
MATH450 Chapter 6: Statistics and Sampling Distributions
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Example 3
Problem
Two airplanes are flying in the same direction in adjacent parallelcorridors. At time t = 0, the first airplane is 10 km ahead of thesecond one.Suppose the speed of the first plane (km/h) is normally distributedwith mean 520 and standard deviation 10 and the second planesspeed, independent of the first, is also normally distributed withmean and standard deviation 500 and 10, respectively.
What is the probability that after 2h of flying, the second planehas not caught up to the first plane?
MATH450 Chapter 6: Statistics and Sampling Distributions