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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 $Y_{n}$?

lhyi561
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    Could you provide exactly what the question states? The question is unclear at present. – LostStatistician18 Apr 13 '22 at 02:02
  • Example 2.3.8 Let $$ Y_{n}= \begin{cases}Y & \text { with probability } 1-p_{n} \ n & \text { with probability } p_{n}\end{cases}, $$ Y may be a constant or a random variable. Then my question is what is the cumulative distribution function of $Y_{n}$?

    I have tried to solve this problem with the total probability rule. That is for every $x\in \mathbb{R}$, $P(Y_{n}<x)=P(Y_{n}<x|Y_{n}=Y)+P(Y_{n}<x|Y_{n}=n).$ But I am not able to determine the outcome of this conditional probability $P(Y_{n}<x|Y_{n}=Y).$

     – lhyi561 Apr 13 '22 at 02:23

1 Answers1

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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 $p_n$.

LostStatistician18
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