Construct an example of sequence of RV Construct a sequence of random variables $X_n$, with below conditions:

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*$X_n \to -1  (Constant RV)$ in probability convergence, as $n\to\infty$.

*$P(X_n\le -1)\to 1$ as $n\to\infty$. (Which is the most confusing part, since convergence type is not indicated. What does that mean?)

*E[$X_n$] diverging to  $\infty$ as  $n\to\infty$.

Let me explain what I think. First condition is pretty straightforward, at least using the definition. Then as I mentioned, I have not an idea about second condition since I did not understand. I can construct that $F_{x_n}(-1)\to 1$ as $n\to\infty$, then use it for convergence in distribution. However, in overall, I am very confused.
 A: For $n\in\mathbb Z_{>0}$ let $X_n\in\mathbb R$ be given by $\mathbb P(X_n=-1)=1-\frac{1}{\sqrt{n}}$ and $\mathbb P(X_n=n)=\frac{1}{\sqrt{n}}$.

*

*We use the definition for convergence in probability from Wikipedia. So, let $\varepsilon>0$ and notice that $\lim_{n\rightarrow\infty}\mathbb P(|X_n-(-1)|>\varepsilon)=\lim_{n\rightarrow\infty}\mathbb P(X_n=n)=\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}=0$. This shows that $X_n$ converges to $-1$ in probability, where as usual $-1$ is a constant random variable in this context.

*We have $\lim_{n\rightarrow\infty}\mathbb P(X_n\le-1)=\lim_{n\rightarrow\infty}(1-\frac{1}{\sqrt{n}})=1$. This is just the standard limit, i.e. for the sequence $a_n=\mathbb P(X_n\le-1)$ of real numbers we have $\lim_{n\rightarrow\infty}a_n=1$.

*We have $\lim_{n\rightarrow\infty}\mathbb E[X_n]=\lim_{n\rightarrow\infty}((1-\frac{1}{\sqrt{n}})\cdot(-1)+\frac{1}{\sqrt{n}}\cdot n)=\lim_{n\rightarrow\infty}(-1+\frac{1}{\sqrt{n}}+\sqrt n)=\infty$.

When confronted with all of the different types of convergence for random variables and distributions, it is hard to properly interpret $\alpha_n\rightarrow\alpha$. The way I solve this problem is that first of all, I check what $\alpha_n$ is. For example, if $\alpha_n$ is a number in $\mathbb R$, like for example $\alpha_n=\mathbb P(X_n\le -1)\in[0,1]$, then I know it's the standard limit. If $\alpha_n:\Omega\rightarrow\mathbb R$ is a random variable, like $\alpha_n=X_n$ in this example, then I expect that the mode of convergence is mentioned somewhere in the text. The same is true if $\alpha_n$ is a probability distribution. To make things even worse, $\alpha_n:\mathcal D\rightarrow\mathbb R$ could also be functions, in which case $\alpha_n\rightarrow\alpha$ usually means pointwise convergence. Truth is, this notation is very overloaded, and should be dealt with carefully.
