Formula Sheet for mathematical statistics

Formula Sheet for mathematical statistics

1. Probability Cheat Sheet 
one page (2 sides) by Peleg Michaeli
peleg.yogiley.org/math/probability/probabilitycs.pdf 

2. Probability and Statistics Cookbook
detailed list by  Matthias Vallentin
http://matthias.vallentin.net/probability-and-statistics-cookbook/cookbook-en.pdf

Note on mathematical statistics: 5.7 Chi-Square Test

1. Mathematical basis of Chi-Square test

• Multivariate normal distribution implies a Chi-Square (n) distribution of Y=Sum( (X_i-mu_i)^2 / sigma_i^2 ). 
• By CLT, the joint pdf of different groups of sample are treated multivariate normal.
• The statistic Q_k-1 = Sum( Y_i^2) = Sum [1..k] ( (X_i-E_i)^2/E_i ) has a Chi-Square(k-1) distribution.
• With the idea of interval  frequency approximation, every distribution can be treated as Multi-binomial.

2. Procedure (the idea of interval  frequency approximation)

• Partition the domain of experiment result into finite mutually disjoint sets A_1, A_2 ... A_n
• Count the number of result in A_i as frequency X_i
• Assign df  (different ways)
• Assign the probability of result in A_i as p_i (different ways)
• Evaluate statistic Q_k-1
• Test

3. Three tests

Test	Goodness of fit	Homogeneity	Independence
example	5.7.1, 5.7.2	5.7.3	5.7.4
H0	Result has the theoretical distribution	Two sets of sample have the same distribution	Two attributions of subjects are independent
Key fact	X_i/n = p_0i	p_1i=p_2i=p_0i=E_i/n	P_ij=Pi*Pj
source of E_i	multinomial model	MLE	MLE
formula of E_i (1<=i<=k)	E_i=n*p_0i	E_i=(X_i1+X_i2)/(n1+n2)	E_ij=(X_i./n) * (X_.j/n)
df	k-1	k-1=2(k-1)-(k-1)	(a-1)(b-1)=(a*b-1)-(a+b-2)
statistic	Sum [i=1..k] ( (X_i-E_i)^2/ E_i )	Sum [j=1,2] [i=1..k]( (X_ij-n_j*E_i)^2/n_j*E_i )	Sum [j=1,a] [i=1..b]( (X_ij-n*E_ij)^2/n*E_ij)
Dataset	x_1, x_2 ... x_n	x_1, x_2,... x_n1 y_1, y_2 ... y_n2	contingency table Xij, 1<=i<=a, 1<=j<=b k=a*b

4. Remarks

• Chi-Square tests are not exact test, but approximate test
• The statistic is based on frequency of result in interval, instead of the result itself
• Make sure E_i > 5 or use Fisher exact test
• Minimum Chi-Square estimation
• MLE based Chi-square tests have greater rejection rate than tests based on Minimum Chi-Square estimator
• Every estimated parameter p0i costs one df

Notes on mathematical statistics: textbook

Here is a serial of notes on my studying on mathematical statistics, specifically on the textbook of
Introduction to Mathematical Statistics,
R.V. Hogg, A Craig and J. W. McKean
6th edition, Pearson.


There was an official solution book pdf file including partial answers to exercises.
However, it can be located and downloaded only after intensive Google search.

I keep my homework and exercises from Chapter 4 to Chapter 7 which can be used as an "AS IS" manual for the book (not only even number questions in official solution, but also some odd numbers questions).

Contact me with the page number and  exercises  index.
I will see what I can do for you.