Table of Contents
What is the conditional distribution of X given that X Y?
For any random variables X and Y, the conditional distribution of Y given X = x specifies how Y varies when X = x. We have already seen instances of conditional distributions when X and Y are independent. In that case, Y varies just as it usually does, regardless of the values of X.
When X and Y are independent then conditional distribution of X given Y becomes?
If X and Y are independent, the conditional pdf of Y given X = x is f(y|x) = f(x, y) fX(x) = fX(x)fY (y) fX(x) = fY (y) regardless of the value of x. Lemma 4.2. 7 Let (X, Y ) be a bivariate random vector with joint pdf or pmf f(x, y).
What is a conditional distribution of a variable?
A conditional distribution is a probability distribution for a sub-population. In other words, it shows the probability that a randomly selected item in a sub-population has a characteristic you’re interested in. This is a regular frequency distribution table.
What does it mean for 2 variables to be independent?
Intuitively, two random variables X and Y are independent if knowing the value of one of them does not change the probabilities for the other one. In other words, if X and Y are independent, we can write P(Y=y|X=x)=P(Y=y), for all x,y.
What is the conditional distribution of Y given x = x?
Note that given that the conditional distribution of Y given X = x is the uniform distribution on the interval ( x 2, 1), we shouldn’t be surprised that the expected value looks like the expected value of a uniform random variable! Let’s take the case where x = 1 2. We previously showed that the conditional distribution of Y given X = 1 2 is
What is the conditional variance of Y?
The conditional variance of Y given X = x is defined as: Although the conditional p.d.f., mean, and variance of X, given that Y = y, is not given, their definitions follow directly from those above with the necessary modifications. Let’s take a look at an example involving continuous random variables.
How to find the conditional mean of Y given x = x?
We can find the conditional mean of Y given X = x just by using the definition in the continuous case. That is: Example: What is the conditional mean of Y given X = x?