January 25, 2007

Statistics Notes

mean   μ = ∑x/n
median (n+1)/2 on ordered sequence
mode   most frequent

variance  = ∑(x-μ)2/n = ∑x2/n - μ2
std-dev = σ = √(variance)

standard score   z = x - μ / σ
[way of comparing values across different data sets ie diff means y std dev's transforms sets of data into new theretical distn with mean of and std dev of 1 standard score = # of std devs from the mean ]

Probability

defn: Event is any occurence that has a probability attached to it.
probability P(A) = n(A)/ n(S)
 n(A)= number of ways of getting an event A
 n(S) = number of possible outcomes
S known as Probability Space or sample Space
A' is known as complemntary event of A (ie A does NOT occur)
P(A') = 1 - P(A)

intersection ∩ (and)
union        ∪ (or)
"given"  |

P(A | B) = probability of event A 'given' event B
P(A | B) = P(A ∩ B)/ P(B)

Law of Total Probability

 P(B) = P(B ∩ A) + P(B ∩ A')
    = P(A)P(B | A) + P(A')P(B | A')
The Law of Total probability is the denominator of Bayes' Theorem

Bayes Theorem

P(A | B) = p(A) x P(B | A)/(P(A) x P(B | A) + P(A') x P(B | A') )

Independent events P(A | B) = P(A)

Discrete Probability Distributions

A random variable use capitals aka Xparticular values of random variables use lowercase aka xTherefore P(X = x) The rpobability that the random variable X takes particular value xvariable is discrete means that it can only take exact values

The Expectation E(X) of a variable X is a bit like the mean, but for probabilyt distributions. To find expectation you multiply each value x by the probability of getting that value and then sum the results

E(X) = ∑xP(X = x)
Tags: statistics mth