Contents
What is stochastic dependency?
“Stochastic” dependence is just the standard definition of lack of independence of probabilities or random variables; see almost any textbook on probability.
Are two variables independent?
The first component is the definition: Two variables are independent when the distribution of one does not depend on the the other. If the probabilities of one variable remains fixed, regardless of whether we condition on another variable, then the two variables are independent.
How do you prove independence of two continuous random variables?
Independence two jointly continuous random variables X and Y are said to be independent if fX,Y (x,y) = fX(x)fY (y) for all x,y. It is easy to show that X and Y are independent iff any event for X and any event for Y are independent, i.e. for any measurable sets A and B P( X ∈ A ∩ Y ∈ B ) = P(X ∈ A)P(Y ∈ B).
How would you know if a variable is independent?
You can tell if two random variables are independent by looking at their individual probabilities. If those probabilities don’t change when the events meet, then those variables are independent. Another way of saying this is that if the two variables are correlated, then they are not independent.
What are the types of stochastic?
Some basic types of stochastic processes include Markov processes, Poisson processes (such as radioactive decay), and time series, with the index variable referring to time. This indexing can be either discrete or continuous, the interest being in the nature of changes of the variables with respect to time.
Can a stochastic random variable be functionally dependent?
However, stochastic independence should not be assumed to imply functional independence; stochastically independent random variables could very well be functionally dependent.
When does the stochastic dependence of X on Y take place?
When stochastic (weak) dependence of y on x takes place a particular value x does not exactly determine a value y but instead affects probability (or probability density or hazard rate) of occuring the random event Y = y.
How is the independence of two stochastic processes defined?
Independencedence of two stochastic processes is a property between two stochastic processes { X t } t ∈ T {displaystyle left{X_{t}right}_{tin {mathcal {T}}}} and { Y t } t ∈ T {displaystyle left{Y_{t}right}_{tin {mathcal {T}}}} that are defined on the same probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},P)} .
How is the concept of Independence used in statistics?
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of the other (equivalently, does not affect the odds).