Probability Distribution, Random Variable, Probability Function, Cumulative Distribution Function

Probability Distribution	Random Variable	Probability Function
Discrete Probability Distribution	Discrete Probability Variable	Probability Mass Function(PMF)	Cumulative Distribution Function(CDF)
Continuous Probability Distribution	Continuous Probability Variable	Probability Density Function(PDF)

 

• Probability Distribution

 

It is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment.

For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails.

 

This is divided into two parts.

- Discrete Probability Distribution

- Continuous Probability Distribution

 

• Feature of Probability Distribution

 

- The probability is between 0 and 1.
- The sum of the sample spaces is 1.
- The probability in any interval is the sum of everything in that interval.

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• Random Variable

 

A variable represented by quantifying the results that may occur in a specific probability experiment.

 

- Discrete Random Variable : The values that can have are countable, such as the number of defective products, the number of accidents.

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• Probability Function

 

It receives the Random Variable as an input and outputs the probability that the value will appear.

- Probability Mass Function(PMF) for Discrete Probability Distribution

- Probability Density Function(PDF) for Continuous Probability Distribution

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• Probability Mass Function(PMF) 

 

- Expression

or

 

The function that presents the probability for a particular value in Discrete Random Variable.

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- Example

 

1) Throwing 3 coins

 

2) 8 times in total

 

3) The number of head : 0, 1, 2, 3 

 

4) Probability Mass Function(PMF) of Throwing 3 coins

Probability Mass Function(PMF)

 

5) Probability Mass Function(PMF) of Y

 

Y : The difference in the number of Head and Tail, it has 1, 3.

TTT : 3-0=3

HTT : 2-1=1

THT : 2-1=1

...

Probability Mass Function(PMF) highlighted by Y

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- Feature of Probability Mass Function(PMF) 

 

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• Cumulative Distribution Function(CDF)

 

 

A function whose result is the probability that the random variable X accumulates from -∞ to a specific point (input).

input : x

output <=x

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• CDF in Discrete Probability Variable

 

- X : A number that comes out of a roll of a dice

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- CDF Graph of X

 

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- CDF Function of X

 

 

[X] is an integer not bigger than x.

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• CDF in Continuous Probability Variable

 

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- Y : The time waiting for the bus that is supposed to be arrived in 10 mins.

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- Y CDF

 

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- CDF Graph of Y

 

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• Fearues of CDF 

 

- The larger the input, the output  is greater than input or equal.

 

if a<b  
Pr(X < a) ≤ Pr(X < b)

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- The smaller the input, the closer the accumulated probability is to zero, and the larger the input, the closer the cdf value is to 1(because it accumulates probabilities for most ranges taken by X).

 

∴

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- Right Continuous

 

 

https://pmxsg.tistory.com/104

https://m.blog.naver.com/stat-mania/221597577634