Statistik task 1

Question 1

How would you define a random phenomenon?

Answer 1

A phenomenon is random if the individual Outcomes are uncertain but there is a regular Distribution of Outcomes in a large number of repetitions.
SO: It is a procedure that has two or more Outcomes, which are not known beforehand

Question 2

What is a probabilisitc Experiment?

& examples

Answer 2

It is an occurence in which the complexity of the Underlying System leads to an Outcome that cannot be known ahead of time. The procedure can be repeated and has a defined set of possible Outcomes (the sample space)

Examples: throwing a dice, tossing a coin

Question 3

How would you definde the sample space in a probabilistic Experiment?

Answer 3

It consist of the complete set of all possible elementary Outcomes of an Experiment: the set of all EOs. When we are drawing random samples, the sample space consists of all possible samples of this size that we could randomly have drawn.

e.g.

Rolling a die once:
sample space = (1,2,3,4,5,6)

Question 4

What are elementary Outcomes?

Answer 4

One possible Outcome of the probability Experiment. When we are drawing a random sample, the sample that was drawn as a whole consists of the elementary Outcome.

E.g:
when drawing one individual (N=1), thousands of Outcomes are possible since we can draw any individual. Therefore, all These individuals constitute an elementary outcome

Question 5

What is an Event?

Answer 5

An Event is a set of one or more elementary Outcomes for one Experiment, a subset of a sample space. An Event with probability 0 never occurs and one with probability 1 occurs on every Trial. The probability of any Event must be between These two possible Outcomes:
0<p></p>

Question 6

Is there a difference between Events and elementary Outcomes?

Answer 6

While an elementary Outcome is one possible Outcome of the sample drawn, meaning that it is Always about the sample itself, an Event is a set of one or more elementary Outcomes. This implies that every elementary Outcome is an Event but not all Event(s) are/is an elementary Outcome.

Question 7

What are disjoint Events?

Answer 7

Two Events A and B are disjoint if there is no elementary Outcomes in common.
Thus, Events are disjoint if they cannot occur simultaneously and that implies that they do not have EO in common.
In mathematical Terms A and B are disjoint if:
P(A|B)=0 and P(A and B)=0

Question 8

What are non-disjoint Events?

Answer 8

Two or more Events are said to be non-disjoint if one or more Events occur at the same time.
For A and B. the probability that either of the Events occur is the sum of the probabilities of the Events, minus the probability that both Events occur is the sum of the probabilities of the Events, minus the probability that both Events occur.

Question 9

offer Examples of disjoint an non-disjoint events

Answer 9

Disjoint:
Football game cannot be Held at the same time as a Rugby game on the same field.
Not possible: getting a red, yellow and green light at a traffic light at the same time

Non-disjoint Events:
six-sided die is rolled once, Event A is rolling an odd number and Event B is rolling a number higher than four,
the Chance that a randomly selected home has a pool or a garage

Question 10

What is a probability model?

Answer 10

Mathematical description of a random phenomenon. It is defined by sample space (S), Events within the sample space and probabilities associated with each Event

P(A) = (Count of Outcomes in A) : Count of Outcomes in
S

Question 11

What is a probability?

Answer 11

The probability of any Outcome of a random phenomenon is the Proportion of times the Outcome would occur in a very Long series of repetitions. That is, probability is a Long-term relative frequency.

Question 12

which values can probabilities have?

Answer 12

1. any probability P(A) is a number between 0 and 1 -> 0<p> P(S) = 1</p>

Question 13

What is the complement rule?

Answer 13

The complement of any Event A is the Event that A does not occur. In order to calculate the unconditional probability that a single Event does not take place, we use the complement rule.

P(Ac) = 1 - P(A)

Question 14

What is a uniform probability model?

Answer 14

In a uniform probability model, each possible Outcome has equal probability. The model can be used to determine probabilites of Events. It holds that:

P(A) = #A : #S

Question 15

How do you determine the probability of an Event using a uniform probability model?

Answer 15

Let us suppose that a stack contains eight tickets numbered 1,1,1,2,2,3,3,3. One ticket will be drawn at random and ist number will be noted. The sample space is S= (1,1,1,2,2,3,3,3)

• P(1) = 3/8
• P(2) = 2/8
• P(3) = 3/8

Question 16

What is a Simple Random Sample?

Answer 16

Simple random sampling (SRS) is a sampling technique where every item in the Population has an even Chance and likelihood of being selected in the sample. The selection of items depends entirely on Chance and therefore, this sampling technique is sometimes known as a method of chances.

Question 17

How are uniform probability models related to simple random sampling or SRS?

Answer 17

In the uniform probability model. each possible Outcome has equal probability and that relates to the SRS beacuse each item of the Population has an equal Chance of being selected

Question 18

What is the Addition rule and when should it be applied?

Answer 18

The Addition rule consists of P(A or B) = P(A)+P(B) - P(A and B)

It should be applied when two Events, A and B are overlapping. P(A and B) refers to the overlap of the two Events.

Question 19

What form does the Addition rule take on in case of disjoint Events?

Answer 19

When two Events, A and B, are mutually exclusive, or disjoint, the probability that A or B will occur is the sum of the probability of each Event.

P(A or B) = P(A) + P(B)

Question 20

What is the multiplication rule and when should it be applied?

Answer 20

The multiplication rule consists of
P(A and B)= P(A) x P(B|A)

If Events A and B come from the same sample space, the probability that both A and B occur is equal to the probability that Event A occurs times the probability that B occurs, given that A has occured. This rule should be used to calculate the unconditional probability that two Events happen simultaneously.

Question 21

What form does the Multiplication rule take on in the case of Independent Events?

Answer 21

If A and B are Independent:
P(A and B)= P(A)xP(B)

When the first Event Happening does not Impact the probability of the seconds, These are called Independent Events. Thus two Events A and B are Independent if knowing that one occurs does not Change the probability that the other occurs.

Question 22

Can disjoint Events be Independent Events?

Answer 22

Do NOT confuse the notions of “Independent Events” and “disjoint Events”.

-> if we know that two Events are disjoint, then if Event A happened, we would know that Event B did not happen. Therefor, Knowledge of A affected our Knowledge of Event B, making them dependent Events.

Question 23

What is a conditional probability?

Answer 23

The conditional probability is the probability that Event B occurs given that Event A has already occured. So, it takes into account some other piece of Information, Knowledge or evidence,

P(A|B) = #(A and B) : #B

Question 24

What is an unconditional probability?

Answer 24

The unconditional probability is the probability that an Event will occur regardless of any past or future occurence of any other Event. So, it does not take into account any other Information, Knowledge or evidence. The unconditional probability is denoted by
P(A) = number of A : total

Question 25

How to use a contingency table to calculate conditional/unconditional probabilities?

Answer 25

1. P(A) -> Marginal:
   Total row or total column / total total
2. P(A and B) -> Joint:
   inside table/ total total
3. P(B|A) -> conditional:
   Inside table/ total row or total column

Question 26

When are two Events statistically Independent?

Answer 26

If one Event does not influence the probability that another Event will take place, These two Events are statistically Independent. 
if P(A) = P(A|B)

Question 27

How can a tree diagram help us to apply the laws of Addition and multiplication in an orderly Fashion?

Answer 27

Tree diagrams Display all the possible Outcomes of an Event. Each branch in a tree diagram represents a possible Outcome. This type of diagra, can be used to find the number of possible Outcomes and calculate the probability of possible Outcomes.

-> when calculating the Overall probabilities, we multiply probabilities along the brances and we add probabilities down columns

Question 28

What do we mean by the term random variable?

Answer 28

A random variable is a quantity that can take on differnt values depending on Chance, namely on the Outcome of a corresponding Experiment.
A value on a random variable constitutes an Event, not an elementary Outcome. When drawing a random sample, it consists of the Features of the persons in the sample

E.g. Experiment with People: Things as age, gender and education Level consist of random variables.

Question 29

Random variables are distributed according to a specific probability Distribution. What does this probability Distribution express?

Answer 29

A probability Distribution is a Display (table or an equation) that links each Outcome of an Experiment with ist probability of occurence

e.g.
Let us suppose the random variable X is defined as the number of heads that result from two coin Flips. Then the table below represents the probability Distribution of the random variable X.
x       P(X)
0       0,25
1        0,50
2       0,20

Question 30

What do we mean by the mean or expected value of a random variable?

Answer 30

The expected value (E) of a random variable X consists of the average value that X would take on if we repeated the Experiment an infinite amount of times. It constitutes out expectation for the single time we actually Conduct the Experiment. It is also called the weighted average.

Question 31

Why is the mean or expected value a weighted average?

Answer 31

because the expected value means a predicted Outcome determined by weighting possible Outcomes by the probability of each Outcome occuring.
In other words, it is a value determined by taking all potential results, multiplying each one by how likely it is to occur and adding them togethe. The sum of These number is the expected value.

Question 32

What do we mean by the term Standard Deviation of a random variable?

Answer 32

The Standard Deviation is a numerical value that consists of the average Degree by which X would deviate from the expected value, if we infinitely repeated the Experiment. In other words, it indicates how widely individuals in a Group vary.

Standard Deviation we are Talking about is the same as the Standard Deviation of a sample- It is acutally the Standard Deviation of a Population:

σ = √ ([Σ(x - u)2]/N)

Question 33

What is the difference between discrete and continous random varibales?

Answer 33

If a variable can take on any value between its Minimum value and its maximum value, it is called a continuoes variable. ( weight is between 75 and 90 kg weight of Person x could take on any value between 75 and 90)
Otherwise, it is called a discrete variable. (Flip a coin -> could be number between 0 and plus infinity. We could not get 2,5 heads)