Maths5

Országok listájaHungaryBudapesti Corvinus EgyetemGazdálkodástudományi KarNemzetközi gazdálkodás (angol nyelven)Mathematics2JegyzetekMaths5

Experiments, sample space, events

5.1

Experiments

Statistical experiments are processes or observations whose outcomes are uncertain. 1. We are tossing a die and observe the number that shows on the top face. 2. We are tossing a die twice and observe the number pair showed on the top face rst and second time. 3. We are tossing a die and then ip a coin as many times as the number showed on the top face. The outcome is the sequence of a number and letters of H and T indicating head and tail respectively. 4. We are tossing a die repeatedly until the number 6 is on the top face. 5. We drop a point onto a circle with unit radius. Additional examples for probability experiments: · We observe the number of cars passing through the intersection of two roads between 10 and 11 o'clock.
· We count the number of phone calls between 8 and 9 arriving at a call center. · We measure the interval of time between two phone calls. · We observe the listing of a stock at closing time. · We measure the awaiting time at the customer service.
5.2 Sample space
The set of all possible outcomes of a statistical experiment is called and represented by the symbol S .

Denition 5.1
the

sample space

The sample space of the previous examples: 1. S = {1, 2, 3, 4, 5, 6} 2. S = {(1, 1), (1, 2), (2, 1), (1, 3), . . . , (6, 6)} 3. S = {1F, 1I, 2F F, 2F I, 2IF, 2II, . . .} (Find the number of elements in this sample space) 4. S is the set of nite sequences of numbers with terms 1,2,3,4,5 and ending with 6. 5. S = {(x, y) : x2 + y 2 1} 1

5.3.

EVENTS

5.3

Events
An

Denition 5.2

event

is a subset of a sample space.

Consider some examples: 1. Let A mean the event, that the number that comes up is even. Then A = {2, 4, 6}. 2. Let A mean the event that the sum of numbers showed on the top face is 7. Then A = {(1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3)}. 3. Let A mean the event that no tail occurs. Then
A = {1H, 2HH, 3HHH, 4HHHH, 5HHHHH, 6HHHHHH}.

4. Let A mean the event that we need at most two tosses to obtain the number 6 on the top face of the die. Then A = {6, 16, 26, 36, 46, 56}. 5. Let A mean the event that the point is closer to the center than 1/2 unit. Then A = {(x, y) : x2 + y 2 < 1/4}.
5.4 Operations with events

The event A S occurs, if the outcome of the experiment is a sample point s S for which s A. Special events are the empty set denoted by having no elements at all, and the whole sample space S . 1. The intersection A B occurs if and only if both A and B occur. Two events A and B are mutually exclusive, if A B = . 2. The union A B occurs if and only if at least one of the events A and B occurs (or both of them occur). 3. The complement of an event A denoted by the symbol A occurs if and only if A does not occur. The occurrence of an event A implies the occurrence of the event B if A B .

Theorem 5.1
1. 2.

(De Morgan formulas)

(A B) = A B (A B) = A B

Similar identities are valid for any number of events.

Some of the outcomes of a statistical experiment are not necessarily observable. For example, when we toss a pair of dice alike, we can not decide which of the outcomes (1, 2) or (2, 1) has occurred. What we can recognize is that the event {(1, 2), (2, 1)} has occurred. Let A denote the set of observable events. It has the following properties:
· S A. · If A A, then A A. · If A1 , A2 , . . . A, then A1 A2 . . . A.

In the following, the pair K = (S, A) will represent an experiment. 2

5.5.

PROBABILITY

5.5

Probability

Let us assume, that an experiment K is repeated n times and we observe the occurrence of an event A A. If the event A has occurred kn times, then the relative frequency of A is given by
kn n

It can be noticed that the relative frequency of the event A is getting closer and closer to a certain number as the n is increasing. This number is considered to be the probability of the event A. We are going to give an axiomatic concept for the probability, from which the above experience can be deduced.

Denition 5.3

Let

K = (S, A) be an experiment. The probability is a function P : A [0, 1]

with the following properties: 1.

P (S) = 1 A1 , A2 , . . . A, are parwise mutually exclusive events, then


2. If

P(
k=1

Ak ) =
k=1

P (Ak )

The triple

(S, A, P ) is called probability space.

The axiomatic concept of probability is due to A. N. Kolmogorov (1933). The subsequent properties of probability are easily deduced from the axioms: 1. For any event A A we have
P (A ) = 1 - P (A)

and therefore P () = 0. 2. If A, B A and A B , then
P (A) P (B)

3. If A, B A, then
P (A B) = P (A) + P (B) - P (A B)

3



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