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For $Y \sim$ Uniform$(-1,1)$ $ Y_{n}= \begin{cases} \text{Y if } > \vert Y \vert \leq 1-\frac{1}{n}\\ \text{n if } \vert Y \vert > >1-\frac{1}{n}\\ \end{cases} $

Prove

  • (a) $ Y_{n}\xrightarrow{L}Y$
  • (b) $ \lim_{n\to \infty}E[Y_n]>E[Y]$

How do I find the CDFs of $Y_n$?

I tried that:

  • (a) if $|Y| \leq 1-1/n$, then $Y_n = Y$, the CDF of $Y_n$ is $F_{Y_n} = \frac{y+1-1/n}{2} $ and the CDF of Y is $F_Y = \frac{y+1}{2}$
    $\lim_{n\to\infty}F_{Y_n} = \frac{y+1}{2}$

    if $|Y| > 1-1/n$, then$Y_n = n$, the CDF of $F_{Y_n} = 1$ if $n > y$
    the CDF of $F_Y = 1$ if y > 1

  • (b) $E[n] = n$, $lim_{n\to\infty}E[n] = n = \infty$

    $E[Y_n] = E[Y] = \theta = 0$

MarianD
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anonyx2
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  • Re your edit: **draw a picture.** The method is illustrated at https://stats.stackexchange.com/questions/138763. – whuber Mar 23 '21 at 18:06

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