Right Continuity of CDF

생성일: 2024년 10월 27일

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질문

Please prove that cumulative distribution function is right-countious

证明累积分布函数的右连续性

首先，我在证明任何一个随机变量 X 的累计分布函数 F 在每个 x ∈ ℝ 上右连续。证明方法包括使用极限公式和平移量 h，让 h 逼近 0，从而证实 F(x+h) 限 h 逼近 0 时等于 F(x)。此外，还利用递减的序列 {x_n} 逼近 x，进一步证明 F 的右连续性。

探讨函数连续性

我正在研究函数F在每个点x的右连续性。通过考虑{xn}趋向x并满足xn > x的序列，目的是证明对于任意ε > 0，有某个δ > 0使得0 < h < δ时，|F(x + h) - F(x)| < ε。

展示连续性

考虑概率测度的性质，展示累积分布函数F(x+) = F(x)。

考察右连续性

我在研究可测集合A_n递减到A时，发现P(∩ A_n)等于P(A_n)的极限。这启示我，类似地，右连续函数F的左极限等于函数值，即F是右连续的。

To prove that a cumulative distribution function (CDF) $F$ is right-continuous at every point $x \in \mathbb{R}$ , we need to show that:

\lim_{h \downarrow 0} F(x + h) = F(x)

Proof:

Definition of CDF:

The cumulative distribution function $F$ of a random variable $X$ is defined as:
$F(x) = P(X \leq x)$
Properties of Probability Measures:
- For any decreasing sequence of measurable sets $\{A_n\}$ such that $A_n \downarrow A$ , the probability satisfies:
  $P\left(\bigcap_{n=1}^\infty A_n\right) = \lim_{n \to \infty} P(A_n)$
- This property follows from the continuity of probability measures from above.
Constructing a Decreasing Sequence:
- Consider a sequence $\{h_n\}$ such that $h_n > 0$ and $h_n \downarrow 0$ .
- Define $x_n = x + h_n$ , so $x_n \downarrow x$ .
Relating Events to the Sequence:
- The events $\{X \leq x_n\}$ form a decreasing sequence of sets:
  $\{X \leq x_{n+1}\} \subseteq \{X \leq x_n\}$
  because $x_{n+1} < x_n$ .
- As $n \to \infty$ , the intersection of these sets approaches $\{X \leq x\}$ :
  $\bigcap_{n=1}^\infty \{X \leq x_n\} = \{X \leq x\}$
Applying the Continuity of Probability Measures:
- Using the property from step 2:
  $P\left(\bigcap_{n=1}^\infty \{X \leq x_n\}\right) = \lim_{n \to \infty} P(X \leq x_n)$
- Therefore:
  $P(X \leq x) = \lim_{n \to \infty} P(X \leq x_n)$
Conclusion:
- This shows that:
  $\lim_{h \downarrow 0} F(x + h) = F(x)$
- Therefore, $F$ is right-continuous at every point $x$ .

Note:

The key idea is that for any $h > 0$ , $F(x + h)$ approaches $F(x)$ from above as $h$ decreases to zero.
This property is a consequence of the continuity of probability measures from above and the non-decreasing nature of CDFs.

Therefore, we have proven that the cumulative distribution function $F$ is right-continuous at every point.

Answer: Because for every real x, F(x + h) decreases to F(x) as h↓0, the CDF F is right-continuous everywhere.

Can you use the same method to prove that when x->negative infinity, the cumulative funtion equals to 0

探讨累积分布函数几点知识

探讨零侧极限

在证明累积分布函数 $F(x)$ 随着 $x$ 趋于负无穷时也趋于0，我们使用概率测度在空集上的连续性。

确定函数特性

函数 $F(x) = P(X \leq x)$ 非递减，且上下界为0和1，概率测度 $P$ 满足 $P(\mathbb{R}) = 1$ 。

分析递减序列

我正在研究递减的 $A_n$ 集合，交集为空集。这表明概率的上连续性，就像以往的证明一样。

你正在研究累积分布函数 (CDF) $F$ 在趋近负无穷时的行为。你通过定义CDF和概率测度的性质，建立一个递减序列并证明 $\lim_{x \to -\infty} F(x) = 0$ 的成立。

定义事件序列

我正在定义事件序列 $A_n = \{X \leq x_n\}$ ，其中 $x_{n+1} < x_n$ 。这个序列是递减的，表示 $X$ 的值随 $n$ 增加而减小。

Yes, we can use a similar method to prove that the cumulative distribution function (CDF) $F$ satisfies:

\lim_{x \to -\infty} F(x) = 0

Proof:

Definition of CDF:

The cumulative distribution function $F$ of a random variable $X$ is defined as:
$F(x) = P(X \leq x)$
Properties of Probability Measures:
- The probability of the empty set is zero:
  $P(\emptyset) = 0$
- For any decreasing sequence of measurable sets $\{A_n\}$ such that $A_n \downarrow A$ , the probability satisfies:
  $P\left(\bigcap_{n=1}^\infty A_n\right) = \lim_{n \to \infty} P(A_n)$
Constructing a Decreasing Sequence:
- Consider a sequence $\{x_n\}$ such that $x_n \to -\infty$ as $n \to \infty$ . For example, let $x_n = -n$ .
- Define the events:
  $A_n = \{X \leq x_n\}$
- Since $x_{n+1} < x_n$ , the sequence $\{A_n\}$ is decreasing:
  $A_{n+1} \subseteq A_n$
Limit of the Events:
- The intersection of all $A_n$ is the empty set:
  $\bigcap_{n=1}^\infty A_n = \emptyset$
  - As $x_n \to -\infty$ , there are no values of $X$ less than all $x_n$ .
Applying the Continuity of Probability Measures:
- Using the property from step 2:
  $P\left(\bigcap_{n=1}^\infty A_n\right) = \lim_{n \to \infty} P(A_n)$
  - Since $\bigcap_{n=1}^\infty A_n = \emptyset$ :
    $P(\emptyset) = 0 = \lim_{n \to \infty} P(X \leq x_n)$
Conclusion:
- Therefore:
  $\lim_{x \to -\infty} F(x) = 0$
  - This shows that as $x$ approaches negative infinity, the cumulative distribution function $F(x)$ approaches zero.

Therefore, using the same method, we have proven that the cumulative distribution function $F$ approaches 0 as $x$ approaches negative infinity.

Answer: Yes; using the same method, we can show that as x approaches negative infinity, F(x) approaches 0.