Lecture 4

Question 1

The sample mean

Answer 1

μ(hat) = 1/M * M Σ i=1 xi

0x1 + 1x2 +…+ / total

Question 2

E(μ(hat)) =

Answer 2

μ

Question 3

var(μ(hat)) =

Answer 3

σ^2/M

Question 4

The central limit theorem

Answer 4

Explains importance of normal pdf in statistics

But still based on asymptotic behaviour of an infinite ensemble of samples that we didn’t actually observe.

Question 5

The bivariate normal distribution p(x,y) =

Answer 5

1/(2πσxσy√(1-ρ^2)) exp[-1/(2(1-ρ^2)) Q(x,y)]

Question 6

where the quadratic form Q(x,y) =

Answer 6

(x-μx/σx)^2 +(x-μy/σy)^2 - 2ρ(x-μx/σx)(x-μy/σy)

Question 7

E(r) =

Answer 7

r^hat = np