Hey I have some questions about this problem:
A bookmaker has placed $n$ bets of the same kind, each bet giving a random payoff $X_i$. It is assumed that the $X_i$ are independently and identically distributed with mean $m$ and variance $σ^2 ∈ (0, ∞)$. Each bet costs $a = m + λσ^2$ for a $λ > 0$.
(a) Let $R$ denote the capital reserve of the betting office and $S_n$ the sum of the payouts $X_i$ . Approximately determine the ruin probability $P(Sn > R + na)$ of the betting office.
(b) According to (a), how large is the ruin probability for $R = 950, σ = 20, λ = 0.001$ and $n = 2500?$ Compare the determined probability with the bound from Chebyshev's inequality.
I have done a) and the first part of b) My question is regarding the bound with the Chebyshev's inequality
The Chebyshev's inequality states $P\left(\left|X-\mu \right|\geq k\right)\leq {\frac {\sigma ^{2}}{k^{2}}}$
What I have done is $P(S_n> R+na)=P(S_n+R(n+\lambda σ^2))=P(Sn-nm> R+n\lambda σ^2)\leq \frac{Var(S_n)}{(R+n\lambda σ^2)^2}= \frac{Var(nσ)}{(R+n\lambda σ^2)^2}$
Now I think I am wrong because i didn't consider the $|.|$ in my formula. In this case, can I ignore it or do I have to change the calculations so that $|.|$ is also included?
If so how should I do? Thanks for any kind of help