We inspect the behavior of the probability that a weighted sum of random variables with log-normal tails is greater than its expected value. Under the right conditions for the weights and the variance being set to 1; we were able to bound a suitable transformation of this probability with the upper bound being a fixed factor of the square root of e above the lower bound. Beyond this, we analyse the conditions on the weights and determine a method for letting the weights be random and give an example.
We end off by extending our result to general variance, where we see that the deviation between the lower and upper bound as well as the domain for the result are dependant on the variance.