What is the expectation of the product of two random variables?
The expected value of the product of two random variables is equal to the product of the expected value, assuming that the variables are independent. Statement: If the two variables X and Y are independent we have that the expectation of XY is equal to the product of the expectation of X and the expectation of Y.
How do you find the expected value of a product of a random variable?
The expected value of the sum of several random variables is equal to the sum of their expectations, e.g., E[X+Y] = E[X]+ E[Y] . On the other hand, the expected value of the product of two random variables is not necessarily the product of the expected values.
What is the expectation of the expectation of a random variable?
Expectations of Random Variables The expected value of a random variable is denoted by E[X]. The expected value can be thought of as the “average” value attained by the random variable; in fact, the expected value of a random variable is also called its mean, in which case we use the notation µX.
What is expectation of product?
– The expectation of the product of X and Y is the product of the individual expectations: E(XY ) = E(X)E(Y ). More generally, this product formula holds for any expectation of a function X times a function of Y . For example, E(X2Y 3) = E(X2)E(Y 3).
Is the product of two Gaussians a Gaussian?
The product of two Gaussian PDFs is proportional to a Gaussian PDF with a mean that is half the coefficient of x in Eq. 5 and a standard deviation that is the square root of half of the denominator i.e. as, due to the presence of the scaling factor, it will not have the correct normalisation.
Is the expectation of a random variable a constant?
The expected value of a constant is just the constant, so for example E(1) = 1. Multiplying a random variable by a constant multiplies the expected value by that constant, so E[2X] = 2E[X]. A useful formula, where a and b are constants, is: E[aX + b] = aE[X] + b.
How do you calculate your expectations?
The basic expected value formula is the probability of an event multiplied by the amount of times the event happens: (P(x) * n). The formula changes slightly according to what kinds of events are happening.
What is the sum of the product of the random variables and its probability?
To find the expected value, E(X), or mean μ of a discrete random variable X, simply multiply each value of the random variable by its probability and add the products. The formula is given as E(X)=μ=∑xP(x).