1. Short review of probabilistic concepts
Probability theory plays a central role in statistics. This lecture gives a short review of the basic concepts of the probability theory.
Contents of this lecture
Basic principles and definitions
Conditional probabilities and independence
Bayes’s theorem and postulate
Random variables and probability distributions
Expectations and moments

2. Random experiment Random experiment satisfies following conditions:
All possible distinct outcomes are known in advance
In any particular experiment outcome is not known in advance
Experiment can be repeated under identical conditions
The outcome space -  is the set of all possible outcomes.
Example 1. Tossing a coin is a random experiment. The outcome space is {H,T} – head and tail.
Example 2. Rolling a die. The outcome space is a set - {1,2,3,4,5,6}
Example 3. Drawing from an urn with N balls, M of them is red and N-M is white. The outcome space is {R,W} – red and white
Example 5. Measuring temperature (in C or in K): What is the outcome space?
Something that might or might not happen depending on the outcome of the experiment is called an event. An event is a subset of the outcome space
Example: Rolling a die. {1,2,3} or {2,4,6}
Example: Measuring temperature in Celsius. Give an example of an event.

3. Classical definition of probability
If all the outcomes are equally likely then the probability of an event A is the number of outcomes in A (M(A)) divided by the number of all outcomes (M):
Example: If a coin is fair then the probability of H is ½ and probability of T is ½
Example: If a die is fair then the probability of {1} is 1/6
If the outcome space is real numbers or are in a space then probability is measured as ratio of the area of an event to that of outcome space:
Where M is the area.
Example: Outcome space is the interval [0,2]. What is the probability of [0,1]?

4. Frequency definition of probability
Since random experiments can be repeated as many times as we wish under identical conditions (in theory) we can measure the relative frequency of occurrences of an event. If the number of trials is m and the number of the occurrences of A is m(A) then according to the frequency definition the probability of A is the limit:
According to the law of large numbers this limit exists. When the number of trials is small then there might be strong fluctuations. As the number of trials increases fluctuations tend to decrease.

5. Other (subjective) definitions of probability
There are other definitions of probability also:
Degree of belief. How much a person believes in occurrence of an event. In that sense one person’s probability would be different from another person’s.
Degree of knowledge. In many cases exact value of a parameter exists but we do not know it. By carrying out experiments we want to find this value. Since experiment is prone to errors it is in general impossible to find the exact value and we assign probability for this. That is the purpose of the most statistical procedures and techniques. According to Jaynes if proper rules are designed then exactly same information would produce exactly same probabilities. (See Jaynes, The Probability theory: Logic of Science). This definition reflects our state of knowledge about parameters and can change as we update our knowledge.

6. Probability axioms
Probability is defined as a function from subsets of outcome space  to the real line - R that satisfies the following conditions:
Non-negativity: P(A)  0
Additivity: if AB= then P(AB) = P(A) + P(B)
Probability of the whole space is 1. P() = 1
All above definitions obey these rules. So any property that can be derived from these axioms is valid for all definitions
Small exercise:
Show that: P( )=0 (Hint:   = )
Show that: 0  P(A)  1 (Hint A and Ã=-A are not intersecting).

7. Example a)
Let us assume that outcome space is a square with sides equal 1 units. Probability of the event A is the area of A. The the probability of either A or B is the sum of areas of A and B. Probability of A and B is zero.
Same as in a). Probability of A is the area of A, probability of B is the area of B. Probability of either A or B is not the sum of he areas of A and B. P(AB)=P(A)+P(B)-P(AB) A B b) A
AB B

8. Conditional probability and independence
Let us consider a case: an event B has occurred or will occur and we want to know what is the probability of A. Knowing B may influence our knowledge about A. Or occurrence of B may influence of the occurrence of A. The probability of A given B is called conditional probability of A given B and is defined as (for P(B)>0):
It is clear that the event B has become new outcome space. Event A and B are called independent if occurrence of B does not influence on probability of A.
It can also be written as:
Note that only one of the above equations is independent.

9. Example
Conditional probability of A given B is the area of AB divided by the area of B. It makes sense since we take it as a fact that B certainly has happened. So probability of A given B will be defined by the set B only.
In some sense we normalise the area of AB by the area of B A
AB B

10. The Law of total probability
In many cases when direct calculation of probability is not known it is easier to divide an event into smaller parts and calculate their probability and then take weighted average of them. This can be done using the law of total probability.
Let B1, B2,,,Bn be partition of , I.e. they are mutually exclusive (BiBj=) and their sum is  (1n Bi= ) then from the axioms of probability:
(Here we do inverse what we did before: remove normalisation of A by the set Bi and then sum over all of them. (P(A|Bi)P(Bi) is probability of A with respect to the original outcome space).
This law is a useful tool to calculate probabilities.
Consider a box with N balls, M of them are red and N-M are white. We make two draws. We don’t know what is the first ball. What is probability of the second ball being red. (Hint: Use partition as ({R1} {W1}). Then use law of total probability for ({R2}. Here subscript shows the first or the second draw.)

11. Bayes’s theorem
Bayes’s theorem is a tool that updates probability of an event in the light of an evidence. It is written in various forms. All they are equivalent. Let us again consider partition of outcome space – B1,B2,,,,Bn so that they are mutually exclusive and sum of them is equal to . Then for one of these events (say j-th event) we can write:
Usually P(Bj|A) is called posterior probability, P(Bj) is prior probability and P(A|Bj) is likelihood. It is widely used in statistical inferences.
Example: A box contains four balls. There are two possibilities: a) all balls are white (B1) b) two white and two red (B2). A ball is drawn and it is white (event A). What is the probability that all balls are white. B1 (all white) and B2 (two white and two red) are two possible outcomes with prior probabilities ½. If B1 is true then probability of A is 1 and if B2 is true then probability of A is ½. Calculate P(B1|A). What is probability P(B2|A)?
Bayes’s postulate: If there is no prior information available then prior probabilities should be assumed to be equal.

12. Random variables
Random variable is a function from outcome space to the real line
X:   R
Example: Consider random experiment of tossing a coin twice. The outcome space is:
={(H,H),(H,T),(T,H),(T,T)})
Define a random variable as
X((T,T)) = 0, X((H,T))=X((T,H)) = 1, X((H,H))=2
Example 2: Rolling a die. Outcome space {1,2,3,4,5,6). Define a random variable
X(j) = j.

13. Probability distribution function
Discrete case (the number of elements in outcome space is finite or countable infinite):
Probability function p assigns for each possible realisation x of a random variable X a probability p(x) = P(X=x). Obviously xp(x) = 1.
Example: The number of heads turning up in two tosses is random variable with probability p(1) = 1/4, p(0) =1/2, p(2) =1/4.
For continuous random variable it is not possible to define probability for each realisation since their probability is usually 0. For them it is easy to define a distribution function:
F(x) = P(Xx)
i.e. probability that X is less than or equal to x. F(x) has the following properties:
1) F(- ) = 0, 2) F(x) is a monotonic and increasing function, 3) F(+ ) = 1.
This function is defined for discrete as well as continuous random variables. If the derivative of F(x) exists then it is called density of probability function – f(x) = dF(x)/dx. Another relation between these two functions is:

14. Cumulative and density of probability distribution
Cumulative probability uniform distribution on the interval [0,1]
Density of probability of uniform distribution on the interval [0,1] b)

15. Joint probability distributions
If there are more than one random variable then their joint probability distribution is defined similarly. For discrete case:
p(x,y) = P((X,Y)=(x,y)) = P(X=x,Y=y)
Then xyp(x,y) = 1, p(x,y)0.
The marginal probability function p(x) is derived by summing over all possible values of y
pX(x) = yp(x,y)
Conditional probability function of X given Y=y is:
p(x|y) = p(x,y)/pY(y)
Definition for the joint probability distribution for continuous random variables is similar.
F(x,y) = P(X  x,Yy). Probability density (f(x,y)) is derivative of the probability function with respect to its arguments. It has properties:
Marginal and conditional probability densities are defined similar to discrete random variables by replacing summation with integration.

16. Joint probability distributions and independence
Random events {X=x} and {Y=y} are independent if
P(X=x, Y=y) = P(X=x)P(Y=y)
The random variables are independent if for all pairs (x,y) this relation holds. It can also be written as
p(x,y) = pX(x)pY(y)
And then p(x|y) = pX(x) and p(y|x) = pY(y)
For continuous random variables definition is analogous. It can be defined by replacing p with f everywhere.
f(x,y) = fX(x)fY(y), f(x|y) = fX(x), f(y|x) = fY(y)
Bayes’s theorem then becomes:
f(x|y) = fX(x) f(y|x)/fY(y)
Where f(x|y) is posterior probability density, f(x) is prior probability density f(y|x) is likelihood of y if x would be observed, f(y) can be considered as a normalisation coefficient.
Usually subscripts X and Y are dropped.

17. Expectation values. Moments
If X is a random variable and h(X) is its function then expectation value (discrete case) is defined as:
E(h(X)) = xh(x)p(x)
If h(x) = x then it is called the first moment. If h(x) = xn then it is called n-th moment. If h(x) = (x-E(X))n then it is called n-th central moment: The second central moment is called variance of the random variable. First moment and second central moment play important role in statistics and they have special symbols
 - is also called as a standard deviation
When there are more than one random variable and their joint probability function is known then their mixed moments also are defined. Most important of them is covariance and correlation:
For continuous random variables expectation values, moments, covariance and correlation are defined similarly by replacing summation with integration. If random variables are independent then their covariance is 0. Reverse is not true in general

18. Examples
Let us take example of tossing a coin. Coin is fair (i.e. probability of head is 0.5 and that of tail is 0.5). Define random variable X(H) = 0, X(T)=1. Then expectation value is:
E(X)=0*P(X=H)+1*P(X=T)=0*0.5+1*0.5=0.5
E(X2)=02*P(X=H)+12*P(X=T)=0*0.5+1*0.5=0.5
E(X-E(X))2=(0-0.5)2*0.5+(1-0.5)2*0.5=0.25* *0.5=0.25
The expectation (first moment) value is 0.5, second moment s 0.5 and standard deviation is 0.5.
Let us take another example. Assume that the density of the probability distribution has the form (it is uniform distribution over the interval [0,1]):
And the random variable is X(x)=x.

19. Further reading
Berthold, M. and Hand, DJ (2003) “Intelligent data analysis”
Feller, W. (1968) An Introduction to Probability Theory and Its Applications: v. 1
Feller, W. (1971) An Introduction to Probability Theory and Its Applications: v. 2
Mardia, KV, Kent, JT and Bibby, JM (2003) “Mutlivariate analysis”
Jaynes, E. (2003) “The probability theory: Logic of science”