*Hi guys, I am working through some questions and was wondering if someone could tell me if I am on the right track as I am not sure, especially with the terms "mathematical expression" and "Write the event". Thank you! *
Let $U_n$ = x + cn − $\sum_{k=1}^n X_k $ be a model for the capital of an insurance company at year n ∈ N, with x,c > 0. Assume ${X_k}_{k≥1}$ is i.i.d. each with distribution exp(1). Let T be the first time the process ${U_n}$ becomes negative.
(a) Give a mathematical expression for the random variable T .
My solution
T = {n: $U_n$ < 0}
b) Write the event {T = 1} in terms of an event involving the random variable $U_1$.
My solution
{T = 1} = $U_1$ = $X_1$ > x + c
c) Calculate P(T = 1).
My solution
P( ${x_1}$ > x + c ) - I am not sure if I am on the right track for this question.