a brief summary of probability theory
Introduction
Factorial:
the product of n consecutive positive integers from n to 1 is n!
n! = n(n-1)(n-2)(n-3)…3.2.1
note: 0! is defined as 1.
Permutation:
each of the ordered subsets which can be formed by selecting some or all of the elements of a set;
the multiplication principle:
if one operation can be performed in m different ways, and when it has been performed in any of these ways, a second operation can then be performed in n different ways, the number of ways of performing the two operations is m x n
eg. if there are 3 different main courses & 4 different desserts, you have a choice of 3×4=12 different two course meals
nPr:
Arrangements in a circle:
Arrangements of n objects in a row, when not all are different:
if p alike of one kind, q alike of another kind, etc..
= n!/(p!q!…);
eg. how many ways can the letters of the word mammal be rearranged to make different words?
Arrangements with restrictions:
Combination:
Mutually exclusive operations:
when the selection of one object eliminates the possibility of it being selected again in that arrangement;
if two operations are mutually exclusive then the no.of arrangements possible with each are added (not multiplied) to obtain the total no. of possible arrangements;
ie. intersection A & B is a null set,
if two or more events cannot occur same time, Pr(A or B) = Pr(A) + Pr(B); (addition principle)
Basic probability definitions
Event: a set of favourable outcomes;
Trial: eg. the tossing of a die;
Sample space: E = all possible outcomes;
Probability of outcomes corresponding to A events:
assume all sample points equally likely,
Pr(A) = [no. outcomes A]/[total no. possible outcomes];
Pr(A or B) = Pr(A) + (Pr(B) - Pr(A&B);
thus, the probability of drawing an Ace or a Heart from a pack of cards = 1/13 + 1/4 - 1/52 = 16/52 = 4/13
Independent events:
Conditional Probability:
Pr(B) given A = Pr(B/A) = Pr(A&B)/Pr(A);
if A,B are independent, then:
Baye's theorem
Statistics:
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2 types of variables:
Population: the group of items/individuals;
Sample range: that part of pop. measured;
Class intervals: subdivisions of the sample range into classes;
Class frequency: no. observations in each class;
Mode: most frequent variable;
Quantile: a value of the variable below which falls a given % of the frequency:
Semi-interquartile range(d) = 0.5(Q3-Q1);
Median: 50% quantile;
Arithmetic mean: the average of a set of observations;
Variance(s2):
Standard deviation(s):
Standard score(z): eliminates scales, but standardises variable wrt mean & s;
Correlation coefficient®:
the degree of assoc. between variables;
r=1, then positively assoc. linear relationship;
r=0, no relationship;
r=-1, then negatively assoc. linear relationship;
Does not allow for non-linear associations & is unduly influenced by extreme observations;
Rank correlation coefficient(r'):
Sampling Distibution:
if x is the mean of the sample, s' is the sd of the pop., n is the no. items in the sample, the standard error of the sampling distribution
= s'/SQR(n);
u (the mean of pop.) almost certainly lies b/n x +/- 3s'/SQR(n);
u(x) = u, s'(x) = s'/SQR(n);
Central Limit Theorem:
Probability Distribution Curves:
-
curve = f(x),
Integral f(x) from -infinity to infinity = 1,
Normal Distribution:
Cauchy Distribution:
Exponential Distribution:
Binomial Distribution:
Hypergeometric Distribution:
when to use:
variables:
N = size of pop.;
n = size of sample;
D = no. of kind A in pop.;
N-D = no of kind B in pop.;
X = no. of kind A in sample;
Pr(X=x) = nCr(D#x).nCr(N-D#n-x)/nCr(N#n)
u = nD/N
s'^2 = nD(1-D/N)(N-n)/[N(N-1)],
if N is very large, & n small, can approx. to binomial distribution;
Poisson distribution:
Hypothesis testing:
Null hypothesis(H0): that there is no effect of one variable on another;
Alternative hypothesis(H1): that there is an effect;
H1 is likely to be true if the results are very unlikely to have been obtained if H0 were true;
Significance level:
Type I error:
Type II error:
b = Pr(accepting H0 when H1 is true);
if p ⇐ a, then reject H0,
if p > a, then accept H0,
Usually a is designated 0.05;
Z-test:
t-test:
Chi-squared test of goodness of fit:
used to test hypothesis concerning the proportions of the pop. in each of the categories;
k = no. of categories in the pop.;
n = no. of random samples from pop.;
Cx = category x;
Ox = observed freq. of sample in Cx;
Ex = expected freq. of sample in Cx if H0 true;
Px = proportion of pop. in Cx;
Ex = nPx, sum(x=1 to k) Px = 1, &o^2 = sum[((Ox-Ex)^2)/Ex], if H0 true, then &o^2 (Chi-squared) small, if H1 true, then &o^2 large;
p = Pr(&^2 > &o^2);
Row by Column contingency tables:
r = no. of rows, c = no. of columns,
Ex = expected freq. assuming independence,
df = (r-1)(c-1),
&o^2 = sum[((Ox-Ex)^2)/Ex], if H0 true, provided Ex > 5 for at least 80% of categ.