Stumped? Calculating Upperbound of Random Variables

[reddavies]

Homework Statement

I have two random variables with two corresponding means and standard deviations. I need to calculate the upperbound that one of the random variables is greater than the other. Any ideas I'm stumped?

Homework Equations

I've used the Markov inequality to calculate the upperbound that a random variable is greater than some given number, but I'm not sure if I can use it here or if I can how to do so.

The Attempt at a Solution

Tried,

P(B - A > 0) <= [E(B) - E(A)] / 0

but this obviously doesn't work. Could it be possible somehow to get the result by subtracting the different means and standard deviations using the Chebyshev inequality?

Say:

P(|B-A|>r) <= (san dev)^2 / r^2

but again r would equal 0.

Thanks for any help you can give!

 

[mighty2000]

Thank you for your post. It seems like you are on the right track with using the Markov and Chebyshev inequalities to solve this problem. However, there are a few things to consider when using these inequalities in this situation.

First, the Markov inequality only applies when the random variable is non-negative. Therefore, you cannot directly apply it to the difference between two random variables.

Second, the Chebyshev inequality gives an upper bound on the probability that a random variable deviates from its mean by a certain amount. In this case, you would need to find the upper bound for the probability that the difference between the two random variables is greater than some number, not just the probability that each individual random variable deviates from its own mean.

One approach you could take is to use the Chebyshev inequality on the difference between the two random variables. This would give you an upper bound on the probability that the difference is greater than some number. Then, you could try to find the upper bound for the probability that one random variable is greater than the other by using the upper bound for the difference and the properties of the normal distribution.

Another approach you could take is to use the properties of the normal distribution directly. Remember that the difference between two normal random variables is also normally distributed, and the mean and standard deviation of the difference is given by the difference of the means and the sum of the variances of the two random variables. You could use this information to find the upper bound for the probability that one random variable is greater than the other.

I hope this helps you in finding a solution to your problem. Good luck!