EDITED:
Tasks 2,3 have been edited as per feedback given in the comments.
my task is as follows:
In a data center there are 2 servers of different types. The probability that a problem occurs in the first year $0.2$ for server 1 and $0.3$ for server 2. Let X be the number of Servers that work without problems in the first year of operation.
- Determine the individual probabilities $P(X = k)$ with $k = 0, 1, 2$.
- Determine the distribution function and its graph.
- Determine the probability that at least one server will work in the first year without problems.
Progress so far:
Task 1: \begin{align*} P(X = 2) &= 0.8 \cdot 0.7 &= 0.56 \\ P(X = 0) &= 0.2 \cdot 0.3 &= 0.06 \\ P(X = 1) &= 1 - 0.56 - 0.06 &= 0.38 \\ \end{align*}
Task 2:
I'm not entirely sure, so I have tried to copy the wiki. Do I simply write $f$ as a piecewise function as follows?
$f_X(x) = \begin{cases} 0.06, \ x = 0 \\ 0.38, \ x = 1 \\ 0.56, \ x = 2 \\ 0, \ x \notin \{0, 1, 2\}\end{cases}$
As for graphing it, the wiki page seems relatively straight forward.
Task 3:
Is there a more elegant/formal/better way of just writing $\Pr(X \geq 1) = P(X = 1) + P(X = 2)$?