Question

Probability conception

Please tell me what is the difference between, symmetric, independent identically distribution and exchangeability

You need to have an example to explain and point the difference between those(Example means some real question and you explain it)

Example,

symmetric has this property but independent identically distribution not......

Answer 1

Symmetric Distribution :

A random variable X is said to be symmetric is if it has it its probability symmetric around its mean

that is

Like cosider the case of normal distribution.

In normal distribution it is symmetric distribution as probablility around mean is equal

i.e.,  

Independent and identically disributed :(iid)

Random variables are said to be iid when all of them belongs from same distribution and are independent

for example

let

which implies that all of them belongs from normal distribution and are independent

i.e,

i.e., we can write the joint distribution of all the random varibles can be written as the product of individual random varibles.

and symmetric and iid are not related to each other.

Exchangeability:

exchangeability is the property that if we exchange , reorder the random variables then what will happen.

like if   are iid random variables then it satisfies the property of exchangibility

for example

because the property of iid

so we can say that iidexchangibility