# Probability’s when each component of the sample is

Probability’s definition

Probability

is the chance that something will happen however it is that events will occur.

Sometimes you can measure a probability with a number like “10%

chance”, or you can use words such as impossible, unlikely, and possible,

even chance, likely and certain.

Mathematics

has many branches one of them is probability which is expressed as a number

between 0 and 1, and that’s calculated by that branch given by the occurrence

of certain event.

For

example the probability of coin toss has only two options either ”tails” or

”heads” this case is considered a probability of one.

Probability

of 0.5 is believed to include same odds if happening or not happening such as

the probability of a coin toss resulting ”heads” or ”tails” but for the

probability of zero is believed to be impossibility, in this case the coin will

land flat without either side facing up that is zero that’s why ”head” or

”tails” must be facing up

It’s

the easiest way can be mathematically

considered as the number of occurrence of specific event divided by the

number of occurrence added to the number of failures of occurrence ( this adds

up to the total of possible outcomes) Pa=Pa/ (Pa+Pb)

When

a single die is thrown , there’s six possible outcomes :1 , 2 , 3 , 4 , 5 , 6

The

probability of one of them is 1/6

Probability

theorems:

Bapat-beg

theorem : In probability theory, the Bapat–Beg theorem provides the joint

probability distribution of order statistics of independent

All components of the sample are gained from the same population and

thus have the same probability

distribution, and the

Bapat–Beg theorem shows the order statistics when each component of the sample

is gained from a various statistical population and therefore has its own probability

distribution.

Markov-krein theorem:

It states that the predicted values of real function of random variables

where only the early moments of random variable are known.

Craps principle theorem:

it’s the theory which talks about event

probabilities below Independent

and identically distributed random variables

trails , as E1 and E2 gives two mutually exclusive events which may happen on a

given trial.

Types of random variables:

A

Random Variable is a set of considerable

significances from a random experiment.

There are two types of random variables:

1-Discrete random variable:

It has limited available significances or

an unlimited series of certain numbers

– X: number of hits on trying 40 free throws.

2-

Continuous random variable:

It

takes all uncountable values in a period of real numbers

– X: the period it

takes for a lamp to burn.

Types

of probability distributions:

1-Geometric

distribution:

On independent Bernoulli trials are

repeated, each with probability p of success, the number of trials X it takes

to get the first success has a geometric distribution.

2-Negative

binomial distribution:

Each with probability P of success,

and X is the trial number when r successes are first accomplished, then X has a

negative binomial distribution. PS: that Geometric (p) = Negative Binomial (p,

1).