What is the expectation of the product of two random variables?

What is the expectation of the product of two random variables?

In general, the expected value of the product of two random variables need not be equal to the product of their expectations. However, this holds when the random variables are independent: Theorem 5 For any two independent random variables, X1 and X2, E[X1 · X2] = E[X1] · E[X2].

Is the expectation of a product the product of the expectations?

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 are the expectations of a function of random variables?

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. Formally, given a set A, an indicator function of a random variable X is defined as, 1A(X) = { 1 if X ∈ A 0 otherwise .

What are the properties of expectation?

The following properties of expectation apply to discrete, continuous, and mixed random variables:

• Indicator function. The expectation of the indicator function is a probability: (5.56)
• Linearity. Expectation is a linear operator: (5.58)
• Nonnegative.
• Symmetry.
• Independence.

Why is expectation linear?

Linearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent. The expected value of a random variable is essentially a weighted average of possible outcomes.

How do you calculate joint expectation?

– 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).

What does a joint probability measure quizlet?

The joint probability of two events equals the probability of the intersection of the two events.

What is expectation 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].

How do you find the expectation of a random variable?

To gain further insights about the behavior of random variables, we first consider their expectation, which is also called mean value or expected value. The definition of expectation follows our intuition. Definition 1 Let X be a random variable and g be any function. 1. If X is discrete, then the expectation of g(X) is defined as, then E[g(X)] = X x∈X

Is expectation a positive linear functional?

jg: (1) In this case, two properties of expectation are immediate: 1. If X(s) for every s2S, then EX 2. 2: Taking these two properties, we say that expectation is a positive linear functional. X is the probability density function for X. Example 1. Flip a biased coin twice and let Xbe the number of heads.

How do you find the expectation of a Bernoulli 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.

How do you find the expectation of a discrete probability distribution?

1. If X is discrete, then the expectation of g(X) is defined as, then E[g(X)] = X x∈X g(x)f(x), where f is the probability mass function of X and X is the support of X. 2. If X is continuous, then the expectation of g(X) is defined as, E[g(X)] = Z ∞ −∞ g(x)f(x) dx, where f is the probability density function of X.