Assignment from it called independent events. Independent events:

Assignment
topic: Probability

Course
title: Statistics

Course
code: SGS116

Lecture
group: C

Submitted
to: DR. Ahmed Refaat

Prepared
by: Mohamed Hassan Kamal /180449

Assignment Topic

Probability

What
is meant by probability?

Mathematically
it could be expressed as the possibility of occurrence of an event divided by
the total number of options you have.

Or it
could be simply: the possibility of occurrence of something.

Theorems
of probability:

There
are several theorems such as addition, multiplication, and Bayes’ theorem.

These
theorems are found because the probability definition cannot be used to find
the the probability of occurrence of at least one of the given events.

Firstly
the addition theorem:

It has many cases such as:    
1.Mutually exclusive

2. Mutually exhaustive

Mutually
exclusive events:

-A
two events are said to be mutually exclusive if they do not have any common
element that is to say if the possibility of event prevents the happening of
the other.

e.g.:
the event of appearance of 2 heads or two tails after tossing a coin

Mutually
exhaustive events:

-A
two events are called mutually exhaustive if the possibility of occurrence of
one of these events is certain (i.e.: P (XUY) =1)

e.g.:
the event of having head or having tail on tossing a coin.

Secondly:
The multiplication theorem:

If X
and Y are two events in the same sample space where P(X) ?0 and P(Y) ?0, then

P (XUY)
=P(X)*P (B?A) =P (B)*P (A?B)

So after
knowing multiplication theory, we have to know that there is a case derived
from it called independent events.

Independent
events: if X and Y are not affected by the occurrence of each other then they
are called independent events.

P (X?Y)
=P(X)*P(Y) (where X and Y are not equal to zero)

N.B:
1.if 3 events are independent then P(A?B?C)=P(A)*P(B)*P(C)

        2.IF X and Y are any two events then
P(AUB)=1-P(A’)-P(B’)

Third theorem: Bayes’ theorem:

This
theorem was named by the scientist Thomas Bayes who was the first one to
provide a formula that allow new evidence to bring up-to-date the beliefs.

And this
formula was developed by Pierre Simon Laplace.

This
theorem is used in the following:

·        
Description
of the probability of an event based on past knowledge related to the
conditions that might have relation with the event.

·        
It is
used in drug testing.

The formula (simplest one):

P (A?B) = (P (A?B)*P (A))/P (B) (A and B are two events ?0)

The types of random variables:

There are mainly two types:

1. Discrete variable.

2. Continuous variable.

The discrete variable:

It is a type of random variable that has either a certain number of
possible values or infinite sequence of countable real numbers.

From its types: Poisson
distribution and binomial distribution.

Value of X

X1

X2

X3

Xi

Probability

P1

P2

P­3

Pi

 

But this type of variables require two things:

1. Every probability must lie between 0&1.

2.?P=1

 

The continuous variable:

It is a variable that takes all values in an interval of numbers.it
is characterized by

Being uncountable, described by density curve.

From its types: normal
distribution, uniform distribution, and exponential distribution.

 

Density
curve   (fig1)

Probability distribution types:

There lots of distributions in probability
the most common are Poisson, binomial, normal, exponential distribution

1.       Binomial distribution: it tests the possibility of event
happening many times exceeding the number of trials and the given probability
in each one.

2.       Poisson distribution: it shows the probability of a given
number of elements in a fixed interval.

3.       Normal distribution: It is the most common used
distribution as it is used in many vital fields such as science and finance.it
is characterized by having mean and standard deviation.

4.       Exponential distribution: it is the distribution that is
related to Poisson distribution as it expresses the time between intervals in Poisson
point process.