cumulative distribution function of special random variable

Question

When I see example 2.3.8 in book Elements of Large-Sample Theory, I have a problem: That is P($Y_{n}$=Y)=1-$p_{n}$ and P($Y_{n}$=n)=$p_{n}$. Obviously, $Y_{n}$ is a piecewise random variable, Then, what is the cumulative distribution function of $Y_{n}$?

Answer 1

Presumably what they mean by this definition is that

$$ Y_n =n1\{B_n = 1\}+ Y 1\{B_n = 0\} $$ where $B_n$ is independent of $Y$ and $P(B_n=1)=p_n = 1-P(B_n=0)$. If so, then by the law of total probability $P(Y_n \le x) = P(Y_n \le x | B_n=0)P(B_n=0)+ P(Y_n \le x | B_n=1)P(B_n=1) =P(Y\le x)(1-p_n) + P(n \le x)p_n.$

Note that this CDF is a mixture of that of $Y$ and a point mass at $n$ of magnitude $p_n$.