Independent Random Variables An independent random variable is a random variable whose occurrence does not depend on the occurrence of any other random vari...
Independent Random Variables
An independent random variable is a random variable whose occurrence does not depend on the occurrence of any other random variable. This means that the probability of an event occurring is independent of the probability of other events occurring.
Joint Distributions
The joint distribution function of two random variables defines the probability of both variables taking a given set of values. The probability of a specific outcome in the joint distribution is given by the product of the individual probabilities of each variable taking that value.
Example:
Suppose you have two random variables, X and Y, that are independent. This means that the probability of X taking a value of 1 and Y taking a value of 2 is independent of the probability of X taking a value of 3.
The joint distribution of X and Y would be a table that shows the probability of each combination of values for X and Y.
Covariance
The covariance of two random variables is a measure of how they are related to each other. The covariance is defined as the average product of the changes in the two variables divided by the square root of the average of the squares of their changes.
Example:
The covariance between X and Y would be a measure of how their values tend to move together. If the covariance is positive, then the variables tend to move in the same direction. If the covariance is negative, then the variables tend to move in opposite directions