2.1. Basic Concepts (Discrete Random Variables)

Connectivity with previous chapters

To simply take the concepts from Chapter 1 (probabilities, conditioning, independence, etc.)
and apply them to random variables rather than events, together with some convenient new notation.

 

numerical values를 가지고 있으면 often useful to assign probabilities to them.

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Lecture note 기준

Random variables

• An assignment of a value (number) to every possible outcome
• Mathematically: A function from the sample space Ω to the real numbers
  • discrete or continuous values
• Can have several random variables defined on the same sample space
  
  
• Notation:
  • random variable X : function Ω => R
  • numerical value x \(\in R\)

 

수업에서 얻은 ideas

* 하나의 sample space가 여러 개의 random variables를 가질 수도 있다.

* random variables 의 function도 random variables이다.

 

 

 

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Book 기준

 

Random variables

Numerical value (or value fo the random variable)

: Given an experiment and the corresponding set of possible outcomes (the sample space), a random variable associates a particular number with each outcome.

 

--> A random variable is a real-valued function of the experimental outcome.

 

 

Visualization of a random variable.

It is a function that assigns a numerical value to each possible outcome of the experiment .

 

An example of a random variable.

The experiment consists of two rolls of a 4-sided die, and the random variable is the maximum of the two rolls.

 

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Main Concepts Related to Random Variables

Starting with a probabilistic model of an experiment:

 

• Random variable: a real-valued function of the outcome of the experiment.
• A function of a random variable defines another random variable.
• We can associate with each random variable certain "averages" of interest, such as the mean and the variance.
• A random variable can be conditioned on an event or on another random variable.
• There is a notion of independence of a random variable from an event or from another random variable.

 

• Discrete random variable: range is either finite or countably infinite.
  • ex) 5 tosses of a coin, the number of heads in the sequence.
• Continuous random uncountably infinite number of values
  • ex) choosing a point a from the interval [- 1. 1]

 

[In this chapter, we focus exclusively on discrete random variables]

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Concepts Related to Discrete Random Variables

Starting with a probabilistic model of an experiment:

 

• A discrete random variable
  : real-valued function of the outcome of the experiment that can take a finite or countably infinite number of values.
• A discrete random variable has an associated probability mass function (PMF)
  : gives the probability of each numerical value that the random variable can take.
• A function of a discrete random variable defines another discrete random variable,
  whose PMF can be obtained from the PMF of the original random variable.

 

 

정리하자면,

Random variable은 sample space Ω to the real numbers를 연결하는 function이다

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References

Bertsekas, D. P., Tsitsiklis, J. N. (2008). Introduction to Probability Second Edition. Athena Scientific.