Probability: Sums and Products of Random Variables

In summary, the distribution functions for the random variables X-Y, XY, and X/Y are determined using the given information about X and Y being uniformly distributed and independent. The distribution functions are found through the process of integrating over the appropriate areas, with the limits of integration determined by the constraints and relationships between the variables.
  • #1
dizzle1518
17
0

Homework Statement


Suppose that X is uniformly distributed on (0,2), Y is uniformly distributed on (0,3), and X and Y are independent. Determine the distribution functions for the following random variables:

a)X-Y
b)XY
c)X/Y

The Attempt at a Solution



ok so we know the density fx=1/2 and fy=1/3. Since they are independent then fxy(xy)=(1/2)*1/3)=1/6. So we will integrate 1/6 over the rectangle with y (height) 3 and x (width) 2. for a) we have P(X-Y<=Z) = P(X-Z<=Y). This is were I get stuck. I am not sure what my limits of integration are. I know that X-Y can be no less than -3 (0-3). and no more than 2 (2-0). How do I go about finding the limits of integration? Same thing with b and c. For example b yield P(XY<=Z) which is equal to P(Z/X<=Y). So that would be the area of the rectangle under the hyperbola Z/X.
 
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  • #2
Your first integral will be [tex]\int_{\{x-y\leq z\}}{f_X(x)f_Y(y)dxdy}[/tex]. Now, try to determine what values x and y can take on (hint: draw a picture!)
 

1. What is the difference between sums and products of random variables?

The sum of two or more random variables is the result of adding their individual outcomes, while the product is the result of multiplying their outcomes. In terms of probability, the sum of random variables follows a different mathematical formula than the product, and the outcomes can vary depending on the distribution of the variables.

2. How do you calculate the mean and variance of sums and products of random variables?

To calculate the mean of a sum of random variables, you simply add the individual means of each variable. For products, you use the property that the mean of the product of two independent variables is equal to the product of their individual means. The variance of a sum of random variables is equal to the sum of their individual variances, while the variance of a product can be calculated using the formula Var(XY) = Var(X)Var(Y) + Var(X)E(Y)^2 + Var(Y)E(X)^2.

3. Can the sum or product of two random variables have a negative outcome?

Yes, both sums and products of random variables can have negative outcomes. This is especially common in distributions with negative values, such as the normal distribution. The outcomes of sums and products are dependent on the individual outcomes of the variables involved.

4. How do you interpret the sum and product of random variables in real-life scenarios?

The sum of random variables is often used to represent the total outcome of a combination of events, such as the total score in a game or the total cost of a shopping trip. The product of random variables is commonly used to represent the probability of multiple independent events occurring simultaneously, such as the probability of flipping two heads in a row. In real-life scenarios, these calculations can help us understand and predict the outcomes of complex events.

5. What are some common applications of sums and products of random variables in science and research?

Sums and products of random variables are commonly used in fields such as statistics, economics, and physics. In statistics, they are used to analyze and interpret data, and to make predictions based on probability. In economics, sums and products of random variables are used to model and predict financial outcomes. In physics, they are used to understand and predict the behavior of complex systems, such as the movement of particles in a gas.

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