We have a program consisting of $n = 100$ modules of code.
$X_i$ = number of errors on the ith module of code
Suppose that $X_i ∼ Poisson(1)$ are i.i.d.
Let $Y = \sum_{i=1}^n X_i $ be the total number of errors.
I need to compute Markov and Chebyshev bounds for $Pr(Y \geq 80)$ but I don't know how.
For instance, I know Markov inequality is:
$$\Pr(X \geq a) \leq \frac{E(X)}{a}$$
And I have seen that for example for $Poisson(1)$ I could replace $E(x)$ for 1, but I'm not sure what I should replace it for in this case, since it is the $E(Y)$.
If the expected value of $X$ is 1, should the expected value of $Y$ be 100?
Cause this doesn't look correct:
$$\Pr(Y \geq 80) \leq \frac{100}{80}$$
Edit
Ok, so since by the comments it seems to be correct. I need to calculate Chebyshev.
For that I need the $Var(Y)$, since $Y$ is a compound poisson distribution I have seen this formula:
$$Var(Y) =(\sigma^2+\mu^2)\lambda$$
And I know $\sigma$ and $\mu$ for $X$ are $\lambda$, so:
$$Var(Y) = 2$$
Then the formula is:
$$\Pr(Y \geq a) \leq \frac{Var(Y)}{(a - E(Y))^2}$$
And I should get:
$$\Pr(Y \geq 80) \leq \frac{2}{400}$$
Is this correct? Should I do anything else on this last formula?