Limit in probability of the mean of squared deviations of a shifted exponential

Let $X_1,X_2,…$ be i.i.d. random variables with common probability density function

$$f(x)=\begin{cases}\frac12e^{-\left(\frac{x-1}2\right)}, & x>1,\ 0, & \text{otherwise.}\end{cases}$$

Define $Y_n=\frac1n\sum_{i=1}^n(X_i-1)^2$, $n=1,2,…$. If $\lim_{n\to\infty}P(|Y_n-k|>\varepsilon)=0$, $\forall\varepsilon>0$, then $k=$