Is conditional expectation random variable?
Conditional expectation, E(X |Y ), is a random variable with randomness inherited from Y , not X.
Is expectation of any random variable is positive?
We will come back to various properties of functions of random variables at the end of this chapter. Using expectation, we can define the moments and other special functions of a random variable. Definition 2 Let X and Y be random variables with their expectations µX = E(X) and µY = E(Y ), and k be a positive integer.
Can a random variable have a negative expectation?
Consider a random variable X, with expectation 1. Now Y:=X−2 is also a random variable and has expectation −1. Of course, the expectation of a non-negative random variable cannot be negative.
Can a random variable be both discrete and continuous?
Random variables can be classified as either discrete (that is, taking any of a specified list of exact values) or as continuous (taking any numerical value in an interval or collection of intervals).
Can a discrete variable be negative?
Your intuition is correct — a discrete variable can take on negative values. The example is just an example: a person can’t have −2 children, but the difference in scores between Home and Away sports teams can be −2 when the Home team is behind by two points.
Which is the conditional expected value of X?
The random variable v(X) is called the conditional expected value of Y given X and is denoted E(Y ∣ X). Intuitively, we treat X as known, and therefore not random, and we then average Y with respect to the probability distribution that remains.
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)].
How is the fundamental property used in conditional expected value?
Moreover the fundamental property can be used as a definition of conditional expected value, regardless of the type of the distribution of \\((X, Y)\\). If you are interested, read the more advanced treatment of conditional expected value. Suppose that \\( X \\) is also real-valued.
What can you say about any non negative random variable?
This identity enables us to extend the definition of integrals of non-negative random variables to integrals of any random variables.) Your condition implies E ( X) = E ( X +), so you may immediately deduce E ( X −) = 0. What can you say about any non-negative random variable whose expectation is zero? – whuber