What is the expectation of one random variable divided over another?

What is the expectation of one random variable divided over another?

What is the expectation of one random variable divided over another (both independent)? Suppose I have X, Y, which are independent random variables. Why is it that E ( X Y) = E ( X) E ( 1 Y)? Also, why is it that E ( X 2 Y 2) = E ( X 2) E ( Y 2)?

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)].

Is the mean and variance of a random pair the unknown ratio?

We are only given the marginal distributions (or rather the mean and variance) of , not the ratio constraint that the data obeyed Simulating from the marginals will result in pairs of that marginally have the “correct” mean and variance, yet many of these pairs may violate the unknown ratio constraint.

Which is the sum of two independent random variables?

Note that the random variables X 1 and X 2 are independent and therefore Y is the sum of independent random variables. Furthermore, we know that: What is the mean of Y, the sum of two independent random variables? And, what is the variance of Y? We can calculate the mean and variance of Y in three different ways.

What are the expectations of functions of independent random?

First note that, since Y is the sum of X 1 and X 2, the support of Y is {0, 1, 2, 3, 4 and 5}. Now, by brute force, we get: The second equality comes from the fact that the only way that Y can equal 0 is if X 1 = 0 and X 2 = 0, and the fourth equality comes from the independence of X 1 and X 2.

Which is the theorem of expectation and independence?

Theorem 2 (Expectation and Independence) Let X and Y be independent random variables. Then, the two random variables are mean independent, which is defined as, E(XY) = E(X)E(Y). More generally, E[g(X)h(Y)] = E[g(X)]E[h(Y)] holds for any function g and h. That is, the independence of two random variables implies that both the covariance and