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Are X and Y independent random variables?
We say that X and Y are independent if P(X=x,Y=y)=P(X=x)P(Y=y), for all x,y. 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.
Is random variable X or Y?
A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables, discrete and continuous.
Are Regressors random variables?
In fact, in a regression model, only the outcome variable is considered a random variable. This means that while we can explain or predict some of its variation, we can’t explain all of it. It is subject to some sort of random process that affects its value in any particular case.
Is the random variable x, y, z mutually independent?
I found the same question ( X,Y,Z are mutually independent random variables. Is X and Y+Z independent? here), but the answer uses characteristic functions and fourier inversion theorem, but this is exercise in chapter long before characteristic functions.
How to find the PDF of random variable z?
Given two independent random variables X and Y, how can I find the PDF of random variable Z = X Y? *If their joint distribution is required, assume that we also have it. The moral: it is not always possible to use this formula. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question.
What is the expectation of a Cauchy random variable?
A Cauchy random variable takes a value in (−∞,∞) with the fol- lowing symmetric and bell-shaped density function. f(x) = 1 π[1+(x−µ)2] The expectation of Bernoulli random variable implies that since an indicator function of a random variable is a Bernoulli random variable, its expectation equals the probability.
Is the expectation of a random variable a linear operator?
In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].