Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that $\mathbf{P}(X > 0) \leq \mathbf{E}(X)$.

For all $N \geq 1$, give an example of random variables satisfying hypotheses $$\mathbb{P}(|X_n| \leq K) = 1, \quad \mathbb{E}[X_n] = 0, \quad \operatorname{Var}(X_n) \leq 1$$ and such that $\mathbb{P}(|S_N| \geq N) \geq 1/2$, where $S_N := X_1 + \cdots + X_N$.