6. Distributions Derived From the Normal Distribution

Question 1

Linear Combination of Normal IID Variables Theorem

Answer 1

-if X1,X2,…Xn are independent random variables with:
Xj~N(µj,σj²)
-then Y=ΣαjXj is also normally distributed with mean Σαjµj and variance Σαjσj²

Question 2

Chi-Squared Distribution

Definition

Answer 2

-a chi-square distribution with r degrees of freedom is a gamma distribution, χ²(r) is the same as Γ(r/2,1/2)

Question 3

Chi-Squared Distribution

Moment Generating Function

Answer 3

M(t) = (1/[1-2t])^(-r/2)

Question 4

Chi-Squared Distribution

χ²(1)

Answer 4

-for the distribution χ²(1):
f(x) = 1/√(2π) * x^(-1/2) * e^(-x/2)
E(X) = 1
Var(X) = 2

Question 5

X~N(0,1), Y=X²

Lemma

Answer 5

-if X~N(0,1), then Y=X²~χ²(1)

Question 6

Y1,Y2,…,Yn iid each χ²(1)

Lemma

Answer 6

-if Y1,Y2,…,Yn are an independent random sample each with a χ²(1) distribution, then:
ΣYj~χ²(r)

Question 7

Two Chi-Squared Random Variables

Corollary

Answer 7

-if Y1~χ²(r) & Y2~χ²(s) and Y1 & Y2 are independent, then:
Y1+Y2 ~ χ²(r+s)

Question 8

X bar

Definition

Answer 8

X_ = 1/n * ΣXi

Question 9

S²

Definition

Answer 9

S² = 1//(n-1) * Σ(Xi-X_)²

Question 10

(X1-X_,…,Xn-X_) and X_

Theorem

Answer 10

if X1,X2,…,Xn are i.i.d. random variables with normal distribution N(µ,σ²), then the vector of random variables:
(X1-X_,X2-X_,…,Xn-X_) and X_ are independent

Question 11

X1,…,Xn i.i.d. ~N(µ,σ²) X_ and S²

Theorem

Answer 11

• if X1,X2,…,Xn are i.i.d. random variables with normal distribution N(µ,σ²), then X_ and S² are independent with distributions:
  i) X_ ~ N(µ,σ²/n)
  ii) (n-1)S²/σ² ~ χ²(n-1)

Question 12

t-Distribution

Definition

Answer 12

-if U~N(0,1) and V~χ²(r) are independent, then:
T = U/√(V/r)
-has a t-distribution with r degrees of freedom, this defines the distribution t(r)

Question 13

t-Distribution

Properties

Answer 13

i) pdf of t(r) :
f(t) = Γ((r+1)/2)/[√(πr)Γ(r/2)] * (1 + t²/r)^(-(r+1)/2), t∈ℝ
ii) if r=1, t(1) is a Cauchy distribution without finite mean or variance
iii) as r->∞, t(r)->N(0,1)

Question 14

X1,X2,…,Xn normal random sample

Theorem

Answer 14

-if X1,X2,…,Xn are a normal random sample then:

√n(X_-µ) / σ ~ t(n-1)