Transformations of Discrete RVs

[dogma]

Hello out there,

I have a question about the transformation of discrete random variables.

I have a joint pdf given by:

[tex]f(x,y)=\frac{(x-y)^2}{7}[/tex] where x = 1, 2 and y = 1, 2, 3

I can easily create a table summarizing the joint pdf of RVs X and Y, f(x,y). I now have a transformation of U = X + Y and V = X - Y.

I'm not quite sure how to go about creating a table to summarize the joint pdf of U and V.

To my feeble mind, it appears that u = 2, 3, 4, 5 and v = -2, -1, 0, 1 (with some numbers for u and v repeated).

How would I go about using u and v and the probabilities from the f(x,y) table to create (transform) the joint pdf, f(u,v)?

I would greatly appreciate someone pointing me in the right direction (i.e. a good, swift kick in the rear). I apologize in advance if some of my terminology is incorrect.

Thanks a bunch,

dogma

 

[dogma]

Well...all's well...figured it out.

 

[tacman]

Hi dogma,

Transformations of discrete random variables can be a bit tricky, but don't worry, I can help guide you in the right direction. First, let's review what a transformation of random variables means. Essentially, it is a way to create new random variables from existing ones by applying a function or formula. In your case, you have two random variables, X and Y, and you want to transform them into two new random variables, U and V, using the formulas U = X + Y and V = X - Y.

To create a table summarizing the joint pdf of U and V, we first need to determine the possible values that U and V can take on. As you correctly stated, for U, the values are 2, 3, 4, and 5, while for V, the values are -2, -1, 0, and 1. These values are obtained by plugging in the different combinations of X and Y from your original joint pdf.

Next, we need to determine the probabilities for each of these values. To do this, we can use the formula for the joint pdf of U and V, which is given by f(u,v) = f(x,y) * |J|, where |J| is the Jacobian of the transformation, which in this case is equal to 1. So, for each possible combination of u and v, we can calculate the corresponding value of x and y and use the probabilities from the original joint pdf to calculate the probability for that combination of u and v.

For example, for u = 2 and v = -2, we have x = 1 and y = 1, so the probability for this combination is f(1,1) = (1-1)^2/7 = 0. Similarly, for u = 4 and v = 1, we have x = 2 and y = 1, so the probability for this combination is f(2,1) = (2-1)^2/7 = 1/7. Continuing this process for all possible combinations of u and v, we can create a table summarizing the joint pdf of U and V.

I hope this helps clarify the process of transforming discrete random variables. Just remember to carefully follow the formulas and pay attention to the different values and probabilities for each combination. Best of luck with your problem!