What is the probability that X and -Y sum to a negative value?
To determine the probability that X and -Y sum to a negative value, we need to have information about the probability distribution of X and Y, as well as any assumptions or conditions that apply.
If we assume that X and Y are independent random variables with known probability distributions, we can calculate the probability of their sum being negative by integrating over the joint distribution.
Let's assume X and Y are continuous random variables, and we denote their probability density functions (PDFs) as fX(x) and fY(y), respectively. The probability that X + (-Y) is negative can be calculated using the joint PDF fXY(x, y) as follows:
1. Start by finding the limits of integration. Since we are interested in the sum being negative, we need to integrate over the area where X + (-Y) < 0.
2. Set up the integral as follows:
P(X + (-Y) < 0) = ∫∫[fXY(x, y) dxdy], where the integral is taken over the appropriate region.
3. Evaluate the integral using the limits of integration derived in step 1.
If X and Y are discrete random variables, a similar approach can be followed by summing the probabilities over the appropriate region instead of integrating.
It's important to note that the specific probability distribution functions for X and Y need to be provided in order to obtain a numerical answer.