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Theme 7. Use of probability in psychological research
1. Introduction.
2. Random variables.
3. Probability function and distribution function.
1. Introduction
Main preliminary concepts
Random experiment: Any operation whose outcome
can not be predicted with certainty
For example, when rolling a dice, RTs when
performing a task, the number of accidents in a given
weekend.
Sample space (E): The set of all possible outcomes of
a random experiment.
For example, if we throw a dice, there are 6 possible
outcomes.
Depending on the number of elements in the sample
space we can distinguish 3 types of sample
spaces:
discrete finite sample space. It consists of a finite
number of elements. (e.g., throwing a dice)
infinite discrete sample space. It consists of a
countable infinite number of elements. (E.g.,
rolling a die until a "6”)
continuous sample space. It consists of an
uncountable infinite number of elements. (E.g.
possible number of achievable points in an
experiment of "throwing an arrow to a target")
Event: It is any subset of a sample space
Event types (according to the number of elements in
the sample space):
Single event (or elemental), that is consisting of a
single element
Compound event, consisting of two or more elements
Sure event, consisting of all elements of the sample
space
Impossible event—that event not composed of any
element of the sample space
Probability:
FORMAL APPROACH
Axioma 1. The probability of the sure event is 1
P( E )  1
Axioma 2. The probability of any event is not negative S
P( S )  0
Axioma 3. The probability of the union of two events (S1 and
S2), mutually exclusive, is the sum of their probabilities
P(S1
S2 )  P(S1 )  P(S2 )
Theorem. The probability of binding a countable infinite set of
mutually exclusive events is equal to the sum of their
probabilities
P(S1
S2
... Sn )  P(S1 )  P(S2 )  ...  P(Sn )
Conditional probability
We called conditional probability of A given / course B
expression
P( A / B)
P( A B)
P( A / B) 
P( B)
Product Theorem
P( A B)  P( B)  P( A / B)
independent events. Two events A and B are statistically
independent if and only if it is verified the following
expression:
P( A B)  P( A)  P( B)
2. Random Variables
A random variable is any function that assigns a real number, and
only one, to each elementary event E; that is, it is any real function
defined on E.
Notation: random varibales are designated by Latin capitals, while the
values attributed to the events are in lowercase letters.
Discrete random variable
One that can only take a finite or countably infinite number of values
Continuous random variable
One that can take an uncountable infinite number of values
3. Probability function
Probability function of X (discrete random variable)
It is the function which assigns to every real number, xi, the
probability that the random variable X assumes that value,
f ( xi )  P( X  xi )
Properties
x1 , x2 ,..., xk
1.
Are the values that can be taken by. X
 f ( x )  P( X  x )  P( X  x )  ...  P( X  x )  1
i
2.
1
2
k
f ( xi )  0
3. Being a <b <c, the event A = {a≤X≤b} and B = {b event <X≤c} are mutually
exclusive:
P(a  X  c)  P(a  X  b)  P(b  X  c)
Function of distibution (discrete random variable)
It is the function which assigns to every real number, the probability that the
random variable X is equal to or smaller than xi
.
F ( xi )  P( X  xi )
Properties
1.
F ()  limF ( xi )  0
xi 
2.
F ()  limF ( xi )  1
3.
F ( xi )
xi 
Is not decrecent
4.
5.
0  F ( xi )  1
F (a  X  b)  F (b)  F (a)
Funtion of density (continuous random variable)
Is that function, f (x) which verifies the following two conditions
1.
f ( x)  0

2.

f ( x)dx  1
The curve, which is the
representation of f (x)
has no points below the
abscissa

The total area under the
curve is 1
f (x) is not a probability: it is a probability density.
Probability density function (continuous random variable). Example
Determine if f (x) is probability density function
f ( x) 
3 x 3  1/ 4
0  x 1
0
otherwise
f(x) will always be at least 0
1
 3x 4   x 
3 1
3
(3
x

1/
4)
dx



 1


0
 4 
4
4
4

0
0
1
So it is
1
Notice that f (x) may be
greater than 1: f (1) = 3'25
Funtion of distribution (continuous random variable)
t is the function which assigns to every real number, x, the probability that
the random variable X is equal to or less than x
x
F ( x)  P( X  x) 

f (t )dt

Properties
1.
F ()  limF ( xi )  0
xi 
2.
F ()  limF ( xi )  1
3.
F ( xi )
xi 
Cannot decrease
4.
5.
0  F ( xi )  1
F (a  X  b)  F (b)  F (a)
discrete vs. continuous random variables
1. In a discrete random variable P (X = x) ≥0 for all x. In a continuous
random variable, P (X = x) = 0 for all values of x.
2. In a discrete random variable , f (x) represents a probability, namely,
P (X = x), and can never be worth more than 1. In a continuous
random variable , f (x) does not represent the probability, but the
probability density (ie, more than one value can).
3. In a discrete random variable , we use points to enter the
probability. In a continuous random variable, we employ
continuous intervals (remember that the probability of each point
is 0).
4. In a discrete random variable , any probability is the sum of
probabilities associated with points. In a continuous random
variable, any probability is a definite integral, associated with an
interval.