Random Variable

• Let the sample space of random experiment be S, X = X(e) is a real-valued single-valued function defined on sample space S. X = X(e) is called a random variable.
• Essentially a function about basic events, independent variable is basic event, dependent variable is function value.

Random Experiment:

Satisfies:

• (1) Repeatability: Experiment can be repeated under same conditions;

• (2) Knowability: Each experiment has more than one possible result, and all possible results of experiment can be clearly identified in advance;

• (3) Uncertainty: Before conducting an experiment, cannot determine which result will appear, but one of the results must appear.

Sample Space:

The set of all basic results of a random experiment is called the sample space. Elements of sample space are called sample points or basic events. That is, sample space is essentially a set, each element is a result of one random experiment.

Sample and Random Variable:

Samples in mathematical statistics have duality, that is, samples can be viewed both as a set of observed values and as random variables.

• First, before sampling. Cannot determine sample’s observed values, so can be viewed as random variables.

• Second, after sample is extracted and observed, sample has specific observed values, so can be viewed as a set of determined values.

Probability Distribution

Let’s look at the simplest coin toss event. Theoretically, probability of heads and tails are both 50%

Try with Code

function flipCoin(){
    for (let index = 0; index < 10; index++) {
        // Round random number
        let randomNum = Math.round(Math.random())
        // If random is 1 then heads
        if(randomNum === 1){
            console.log('Heads')
        }else{
            console.log('Tails')
        }
    }

}

flipCoin()

Try 10 times result:

Try 1000 times

• More sampling times in statistics, closer to theoretical situation

• Probability distribution actually describes probability law of random variables.

Discrete Distribution Models

Bernoulli Distribution

This is distribution of a single random variable, and this variable only has two values, 0 or 1.

or

Example:

Assume you want to have children, probability of boy is p, probability of girl is 1-p

Bernoulli experiment: Have one child

Bernoulli distribution: Have one child, probability of boy is p, probability of girl is 1-p, this is Bernoulli distribution

Categorical Distribution (also called Multinoulli Distribution)

It describes a single random variable with k different states. Here k is a finite value, if k is 2, then categorical distribution becomes Bernoulli distribution. I’ve listed the formula and diagram of this distribution.

Normal Distribution

Formula:

In this formula there are two parameters, μ represents mean, σ represents variance.

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