Right Continuity of CDF

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

Proof:

1. Definition of CDF:

   The cumulative distribution function $F$ of a random variable $X$ is defined as:

   $$F(x) = P(X \leq x)$$

2. 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.

3. 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 $x_n \downarrow x$.

4. 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 $\{X \leq x\}$:

     $$\bigcap_{n=1}^\infty \{X \leq x_n\} = \{X \leq x\}$$

5. 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:

     $$P(X \leq x) = \lim_{n \to \infty} P(X \leq x_n)$$

6. 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.

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

Proof:

1. Definition of CDF:

   The cumulative distribution function $F$ of a random variable $X$ is defined as:

   $$F(x) = P(X \leq x)$$

2. 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:

     $$P\left(\bigcap_{n=1}^\infty A_n\right) = \lim_{n \to \infty} P(A_n)$$

3. 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:

     $$A_{n+1} \subseteq A_n$$

4. 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 $x_n$.

5. 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 $\bigcap_{n=1}^\infty A_n = \emptyset$:

       $$P(\emptyset) = 0 = \lim_{n \to \infty} P(X \leq x_n)$$

6. 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.