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.