Contents
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
When does the expectation of x equal the expected value?
In other words, if X and Y are random variables that take different values with probability zero, then the expectation of X will equal the expectation of Y. . In particular, for a random variable . A well defined expectation implies that there is one number, or rather, one constant that defines the expected value.
What is the definition of independence in statistics?
Independence (probability theory) 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,
Which is an example of the property of expectation?
Before looking at more examples, we would like to talk about an important property of expectation, which is linearity. Note that if X is a random variable, any function of X is also a random variable, so we can talk about its expected value. For example, if Y = aX + b, we can talk about EY = E[aX + b].
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)].
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.