Probability of mutually independent random variables I have been doing an exam review and this is one of the problems I could not solve.

Let $X_1, X_2, ..., X_n$ be a sequence of mutually independent random variables.
For each $i$ with $1 \leq i \leq n$,


*

*the variable $X$, is equal to either $0$ or $n+1$

*$E(X_i) = 1$

Determine $Pr(X_1+ X_2 + ... + X_n  \leq n)$


If $X$ could be $0$ or $n+1$ you can call $p$ to the probability of being $0$ and obviously $1-p$ is the probability of being $n+1$
How the expectation is equal to $1$:
$$E(X)=p\cdot 0 + (1-p)(n+1)=1$$
$$p=\frac{n}{n+1}$$
For one side:
$$\sum_{i=0}^n X_i \leq n \Rightarrow X_i\leq n , \forall i=1..n\Rightarrow X_i=0, \forall i=1..n$$
For other:
$$X_i=0, \forall i=1..n \Rightarrow\sum_{i=0}^n X_i=0 \leq n$$
Then:
$$\sum_{i=0}^n X_i \leq n \iff X_i=0, \forall i=1..n$$
Then:
$$P\left(\sum_{i=0}^n X_i \leq n\right)=P(X_i=0, \forall i=1..n)=p^n=\left(\frac{n}{n+1}\right)^n$$
The last condition is accomplished via the independence of the variables.
