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Is there proof for joint probability density of independent variables?
(i.e. joint probability density of independent random variables is equal to the product of marginal densities) ? If so, what is the proof behind this theorem (or should the statement be treated as the definition of independence, rather than a theorem)? It’s not a homework, I am asking because I am curious what the proof is.
When are X and Y independent of each other?
X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) i.e. Proof: • If X and Y are independent random variables and Z =g(X), W = h(Y) then Z, W are also independent. ,X Y ( , )= ( ) X Y (F x y F x F y )
How are random variables x and Y independent?
Random variables X and Y are independent if their joint distribution function factors into the product of their marginal distribution functions • Theorem. Suppose X and Y are jointly continuous random variables. X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) i.e.
Which is an example of a joint CDF?
We have already seen the joint CDF for discrete random variables. The joint CDF has the same definition for continuous random variables. It also satisfies the same properties. F X Y ( x, y) = P ( X ≤ x, Y ≤ y).
How to calculate variance of product of 2 independent random vector-cross?
Here’s a matrix-notation version that might be more convenient to work with than the plug-and-chug required by Dilip’s answer . Let μX: = E[X], ΣX: = Var[X], and similarly for μY, ΣY .
How to calculate the variance of a random variable?
This answer supposes that XTY (where X and Y are n × 1 vectors) is a 1 × 1 vector or scalar ∑iXiYi and so we need to consider the variance of a single random variable that is this sum of products. Since X and Y are independent random vectors, we note that X1, Y1 are independent random variables as are X2, Y2.