Summary
Summary probability and statistics
- Course
- Institution
Summary of the first years course probability and statistics
[Show more]
Seller
Follow
jochemnauta
Content preview
Contents1 Chapter 4 2
1.1 Discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 more properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Empirical distribution functions . . . . . . . . . . . . . . . . . . . 2
2 Chapter 5 3
2.1 Conditional expectation . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Bivariate Normal distribution . . . . . . . . . . . . . . . . . . . . 4
3 Chapter 6 4
3.1 CDF-approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 pdf-approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 mgf approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Order statistics and their distribution . . . . . . . . . . . . . . . 5
3.5 Jensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 chapter 7 6
4.1 Convergence in terms of densities . . . . . . . . . . . . . . . . . . 6
4.2 Weak law of large numbers (WLLN) . . . . . . . . . . . . . . . . 6
4.3 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . 6
4.4 Delta Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 Chapter 8 7
5.1 Chi-square distribution . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2 F- distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Chapter 9 8
6.1 Method of moments . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.2 Method of Maximum likelihood . . . . . . . . . . . . . . . . . . . 9
6.3 quality of estimators . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.4 Cramer-Rao inequality . . . . . . . . . . . . . . . . . . . . . . . . 10
1
, 1 Chapter 4
The k-dimensional discrete random vector (X1 , ...., Xk ) has a density function
defined by:
f (x1 , ......, xk ) = P (X1 = x1 , X2 = x2 , ..., Xk = xk )
We will discuss a few multivariate distributions:
The mutinomial distribution, the extension of the binomial distribution:
xk+1
n!
P (X1 = x1 , X2 = x2 , ..., Xk = xk ) = x1 !∗x2 !∗...x∗k!∗x∗k+1! px1 1 px2 2 ∗ ...... ∗ pk+1
1.1 Discrete case
P P
Theorem: x1 .... xk f (x1 , ..., xk ) = 1 P
Definition for marginal densities: fX2 (x2 ) = allx1 f (x1 , x2 )
Definition: F (x1 , ..., xk ) = P (X1 ≤ x1 , X2 ≤ x2 , ....., Xk ≤ xk
We also can have that:
P (x11 < X1 ≤ x12 , x21 < X2 ≤ x2 2)
= F (x12 , x22 ) − F (x12 , x21 ) − F (x11 , x22 + F (x11 , x21 ), is bigger or equal to zero,
obviously since it is a probability.
1.2 Continuous case
R xk R x1
F (x1 , ...., xk ) = −∞ ..... −∞ f (t1 , ..., tk )dt1 ....dtk
Hence it also holds that:
δk
f (x1 , ....., xk ) = δx .....δx1
F (x1 , ...., xk )
1
Additionally, taking the integral of all xi from −∞ to ∞ is equal to 1.
1.3 more properties
R R
P (X ∈ Region ) = Region f (x1 , x2 )dx1 dx2
For f (and F) we have that k random variables (X1 , ..., Xk ) are said to be inde-
pendent if f (x1 , ..., xk ) = fX1 (x1 ) ∗ ..... ∗ fXk (xk )
Conditional density is: f (x2 |x1 ) = ff(x 1 ,x2 )
X1 (x1 )
For f is a uni variate density function and (X1 , ..., Xn ) an n-dimensional random
vector, if g(x1 , ...., xn ) = f (x1 ) ∗ .... ∗ f (xn ) holds for the joint density g, then
X1 , ..., Xn is said to be a random sample
1.4 Empirical distribution functions
X1:n ≤ X2:nP≤ ... ≤ Xn:n The empirical distribution function is defined by:
n
Fn (x) = n1 i=1 1(−∞,x) (Xi )
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jochemnauta. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $5.88. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
52510 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now