Calculating Conditional Expectation of X,Y,Z: Triangle Inequality

In summary, we discuss calculating the characteristics function and density of a Gaussian (T) with standard normal variables (X,Y,Z). We also explore two methods for calculating E[Max(X,Y)] for N(0,1) variables (X and Y). Finally, we prove the triangle inequality for random variables X and Y with second moments.
  • #1
kliwin
1
0
1.
Let T = (X,Y,Z) be a Gaussian for which X,Y,Z for which X, Y, Z are standard normals, such that E[XY] = E[YZ] = E[XZ] = 1/2.

A) Calculate the characteristics function Φ_T(u,v,w) of T.
B) Calculate the density of T.

2.
Let X and Y be N(0,1) (standard normals), not necessarily independent. Calculate E[Max(X,Y)] using two different ways:

A) Use the joint density of X and Y and use the fact that for two numbers x and y, max(x,y) = x if x > y and y if y > x.

B) Use the change of variables x = x and y = ρx = u*sqrt(1-ρ^2) in your integral.

3.
Given random variables X and Y whose second moments exit, prove the triangle inequality

E[(X+Y)^2]^(1/2) <= E[X^2]^(1/2) + E[Y^2]^(1/2)

Help guys, this is for my Advanced Probability Course and I am stuck on it. :/
 
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  • #2
Proof: Let X,Y be random variables with second moments. We want to prove the triangle inequality E[(X+Y)^2]^(1/2) <= E[X^2]^(1/2) + E[Y^2]^(1/2).We can rewrite the expression as E[(X+Y)^2] <= (E[X^2] + E[Y^2])We can expand the left-hand side of the inequality as follows: E[(X+Y)^2] = E[X^2 + 2XY + Y^2] Using the linearity of expectation we have: E[(X+Y)^2] = E[X^2] + E[2XY] + E[Y^2] Now, using the Cauchy-Schwarz inequality, we have: E[2XY] <= sqrt(E[X^2]*E[Y^2]) Substituting this in the equation for E[(X+Y)^2], we get: E[(X+Y)^2] <= E[X^2] + sqrt(E[X^2]*E[Y^2]) + E[Y^2] Simplifying gives us: E[(X+Y)^2] <= (E[X^2] + E[Y^2]) + sqrt(E[X^2]*E[Y^2]) Since the square root of a positive number is greater than or equal to zero, we have: E[(X+Y)^2] <= (E[X^2] + E[Y^2]) This proves the triangle inequality.
 

1. What is conditional expectation?

Conditional expectation is a statistical concept that measures the expected value of a random variable given the knowledge of another related random variable. In other words, it calculates the average value of a random variable based on the information provided by another random variable.

2. How do you calculate conditional expectation?

The formula for calculating conditional expectation is E(X|Y) = ∑x P(X=x|Y) * x, where X and Y are random variables, P is the probability function, and x is the value of the random variable X. In simpler terms, it is the sum of the product of each possible value of X and the probability of that value occurring given the value of Y.

3. What is the Triangle Inequality?

The Triangle Inequality is a mathematical principle that states that the sum of any two sides of a triangle must be greater than the third side. In other words, the shortest distance between two points is a straight line.

4. How is the Triangle Inequality used in calculating conditional expectation?

In the context of calculating conditional expectation, the Triangle Inequality is used to prove that the expected value of the sum of two random variables is greater than or equal to the sum of their individual expected values. This is a key step in the proof of the formula for calculating conditional expectation.

5. What are some real-world applications of calculating conditional expectation using the Triangle Inequality?

Calculating conditional expectation using the Triangle Inequality has many practical applications, such as in finance and economics. For example, it can be used to predict stock prices or to determine the expected return on an investment based on various market conditions. It can also be used in risk analysis and decision-making processes to evaluate potential outcomes based on different scenarios.

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