What is expectation of a discrete random variable?
For a discrete random variable the expected value is calculated by summing the product of the value of the random variable and its associated probability, taken over all of the values of the random variable.
What are indicator random variables?
An indicator random variable is a special kind of random variable associated with the occurence of an event. The indicator random variable IA associated with event A has value 1 if event A occurs and has value 0 otherwise. In other words, IA maps all outcomes in the set A to 1 and all outcomes outside A to 0.
Which is the formula for a conditional expectation?
Of course it is given by fXjY (xjy) = P(X = x;Y = y) P(Y = y) = fX;Y (x;y) fY (y) This looks identical to the formula in the continuous case, but it is really a di erent formula. In the above fX;Y and fY are pmf’s; in the continuous case they are pdf’s.
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.
Is the expectation of X independent of y 1 and y 2?
Literally, the expectation of X given the values of both Y 1 and Y 2. “The average height (X) of a 12 year old (Y1) male (Y2)” is an example. Is E [ X | Y 1, Y 2] equivalent to E [ X | Y 1] ⋅ E [ X | Y 2]? No. Consider the trivial case where X is independent of both Y 1 and Y 2.