What is the Expected Value of (X/Y)?

In summary, the conversation discusses calculating the expected value of the ratio of two discrete random variables with a given joint probability density function. The attempt at a solution involves using integrals, but it is incorrect as integrals are only used with continuous random variables. A better approach would be to calculate the expected value by finding the ratio for each state and taking a weighted sum.
  • #1
Samwise_geegee
6
0

Homework Statement


Let x and y be discrete random variables with joint probability density function

P(X,Y)= 2X-Y+1/9 for x=1,2 and y=1,2
0 Otherwise

Calculate E[X/Y]


Homework Equations



E[XY]= ∫∫XYP(X,Y)dxdy



The Attempt at a Solution



I can't find a property of Expected value to make E[X/Y] solvable. This is my best guess.
E[X/Y]= ∫∫X*1/YP(X,1/Y)dxdy

E[X/Y]=∫∫X*1/Y*(2X-1/Y+1/9)dxdy from 1 to 2 on the first integral and 1 to 2 on the second integral
 
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  • #2
looks like you have a discrete probability function there.
there are only 4 states, so why not work out X/Y for each state and find the expectation by weighted sum?
 
  • #3
I'm just not sure how to calculate E(X/Y). Is what I wrote right?
 
  • #4
You can answer that one yourself!

Consider: what you wrote involves some integrals.
Are integrals associated with continuous or discrete probability functions?
What kind do you have?
 
  • #5
Samwise_geegee said:
I'm just not sure how to calculate E(X/Y). Is what I wrote right?

What you wrote is wrong. You seem to be suffering from the disease of writing formulas without knowing what they mean or when they should be used, and the prognosis of that disease is not good. Integrals are used with continuous random variables having probability density functions. Do you see any such random variables in your problem?
 

1. What is the expected value of (X/Y)?

The expected value of (X/Y) is a measure of the average outcome of a random variable (X) divided by another random variable (Y). It represents the theoretical long-term average that would occur if the experiment is repeated an infinite number of times.

2. How is the expected value of (X/Y) calculated?

The expected value of (X/Y) can be calculated by taking the sum of all possible outcomes of (X/Y) multiplied by their respective probabilities. This can be represented mathematically as E(X/Y) = Σ(x/y)*P(x/y), where x/y represents each possible outcome and P(x/y) represents the probability of that outcome occurring.

3. What does the expected value of (X/Y) tell us?

The expected value of (X/Y) tells us the average outcome that we can expect from dividing one random variable by another. It is an important concept in probability and statistics as it helps us understand the likelihood of different outcomes and make informed decisions.

4. How does the expected value of (X/Y) relate to risk and uncertainty?

The expected value of (X/Y) is a useful tool for assessing risk and uncertainty. It allows us to calculate the potential outcomes of a situation and weigh them against their respective probabilities. This helps us make more informed decisions by considering the potential risks and uncertainties involved.

5. Can the expected value of (X/Y) be negative?

Yes, the expected value of (X/Y) can be negative. This would occur when the potential outcomes of (X/Y) are negative and their probabilities are high enough to outweigh any positive outcomes. It is important to consider both positive and negative outcomes when calculating the expected value of (X/Y) to get a comprehensive understanding of the situation.

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