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For the given Random Variable X, Cumulative Distributive Function is defined as below:

$$F(x) = \begin{cases} 0, & \text{$x \le 0$} \\ x^2/8, &\text{$0\le x \lt 2$}\\ 1,& \text{$x \ge 2$} \end{cases} $$

And Let the two events $C_k, D_k$ be

$$C_k = \{x \mid 1/k \le x \le 2-1/k\}$$ and $$D_k = \{x \mid 2-1/k \lt x \lt 2+1/k\}$$ where $k \in \Bbb N$

(a) $P(X\in \lim\limits_{k \to \infty}C_k) = P(X \in \{0 \le k \le 2\})=1\begin{align}\end{align}$

(b) $P(X\in \lim\limits_{k \to \infty}D_k) = P(X \in \{2 \lt k \lt 2\})=P(X \in \emptyset)= 0\begin{align}\end{align}$

Is my above reasoning (a), (b) correct?

Beverlie
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1 Answers1

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$\require{begingroup}\begingroup \newcommand{\P}{\operatorname{\sf P}}$Mostly. There is a step discontinuity in the CDF (a massive point) at $2$ , so whether or not the limit point is interior to interval is important.

  • Because $C_k=[1/k;2-1/k]$ then for no $k\in\Bbb N$ is $0\in C_k$ nor is $2\in C_k$, so $\lim_{k\to\infty}C_k=(0;2)$. $$\P\big(X\in(0;2)\big)=1/2$$

  • Since $D_k=(2-1/k;2+1/k)$, then for all $k\in\Bbb N$ we have $2\in D_k$, so $\lim_{k\to\infty}D_k=\{2\}$.$$\P\big(X\in\{2\}\big)=1/2$$

$\endgroup$

Graham Kemp
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    First, I want ask you, isn't it 1/2 and 1/2 for each case? since x^2/8 not x^3/8. Second, I've never thought about but the confinement of k in $\Bbb N$ has made the problem. How could I be or well-knowledged to deal with difference between infinite case of Natural numbers and Real numbers? In my head rightnow, there's no 'understand' but only 'In case of Natural numbers only, the given 1\k never could be 0' – Beverlie Jun 24 '17 at 06:54
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    Good catch. Yes, there is a step discontinuity in the CDF, so whether or not $2$ is included is important. – Graham Kemp Jun 24 '17 at 07:05
  • if the k is not natural number, but real numbers, then it changes the result correspoding to OP? – Beverlie Jun 24 '17 at 07:15
  • No. Not at all. – Graham Kemp Jun 24 '17 at 07:31
  • I see. So equal sign which denotes convergence doesn't logically applicable in this problem Set's sense – Beverlie Jun 24 '17 at 07:34