Chapter 6: Statistics and Sampling Distributionsvucdinh.github.io/Files/lecture05.pdf · 2020. 12. 15. · Chapter 6: Statistics and Sampling Distributions Week 4 Chapter 7: Point

Chapter 6: Statistics and Sampling Distributions

MATH450

September 12th, 2017

MATH450 Chapter 6: Statistics and Sampling Distributions

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


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


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


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


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?)


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).



Property

Two distributions have the same pdf if and only if they have thesame 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)


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.



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))


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


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


Example 1

Problem

Assume that

X1 ∼ N (10, 9) and X2 ∼ N (30, 16)

are independent.

What is the distribution of X1 − X2?


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?


How to do computations with normal random variables?

Reading: 4.3


N (µ, σ2)


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


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


Φ(z)

Φ(z) = P(Z ≤ z) =∫ z−∞

f (y , 0, 1) dy


Φ(z)


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 )


Shifting and scaling normal random variables


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?


Documents

Chapter 6: Statistics and Sampling Distributionsvucdinh.github.io/Files/lecture05.pdf · 2020. 12. 15. · Chapter 6: Statistics and Sampling Distributions Week 4 Chapter 7: Point