Question

Prove not always uncolreated randon variable are independent

Answer 1

Step-by-step explanation:

If two random variables X and Y are independent, then they are uncorrelated. ... Uncorrelated means that their correlation is 0, or, equivalently, that the covariance between them is 0. Therefore, we want to show that for two given (but unknown) random variables that are independent, then the covariance between them is 0

Answer 2

Step-by-step explanation:

Mathematics is known for its resolute commitment to precision in definitions and statements. However, when words are pulled from the English language and given rigid mathematical definitions, the connotations and colloquial use outside of mathematics still remain. This can lead to immutable mathematical terms being used interchangeably, even though the mathematical definitions are not equivalent. This occurs frequently in probability and statistics, particularly with the notion of uncorrelated and independent. We will focus this post on the exact meaning of both of these words, and how they are related but not equivalent.

Independence

First, we will give the formal definition of independence:

Definition (Independence of Random Variables).

Two random variables XX and YY are independent if the joint probability distribution P(X, Y)P(X,Y) can be written as the product of the two individual distributions. That is,

P(X, Y) = P(X)P(Y)

P(X,Y)=P(X)P(Y)

Essentially this means that the joint probability of the random variables XX and YY together are actually separable into the product of their individual probabilities. Here are some other equivalent definitions:

P(X \cap Y) = P(X)P(Y)P(X∩Y)=P(X)P(Y)

P(X|Y) = P(X)P(X∣Y)=P(X) and P(Y|X) = P(Y)P(Y∣X)=P(Y)

This first alternative definition states that the probability of any outcome of XX and any outcome of YY occurring simultaneously is the product of those individual probabilities.

For example, suppose the probability that you will put ham and cheese on your sandwich is P(H \cap C) = 1/3P(H∩C)=1/3. The probability that ham is on your sandwich (with or without any other toppings) is P(H) = 1/2P(H)=1/2 and the probability that cheese is on your sandwich (again, with or without ham or other goodies) is P(C) = 1/2P(C)=1/2. If ham and cheese were independent sandwich fixings, then P(H\cap C) = P(H)P(C)P(H∩C)=P(H)P(C), but

P(H\cap C)= 1/3 \neq 1/4 = P(H)P(C)

P(H∩C)=1/3



=1/4=P(H)P(C)

Thus, ham and cheese are not independent sandwich fixings. This leads us into the next equivalent definition:

Two random variables are independent if P(X|Y) = P(X)P(X∣Y)=P(X) and P(Y|X) = P(Y)P(Y∣X)=P(Y).

The vertical bar is a conditional probability. P(X|Y)P(X∣Y) reads “probability of XX given YY“, and is the probability XX will have any outcome xx given that the random variable YY has occurred.