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Let X be a continous random variable with pdf fX(x) and cdf FX(x). Let A be a subset of the real line. Let I_A(x) be the indicator function for A. Find an expression for the cdf of Y.

To solve this problem, we need to find the cumulative distribution function (CDF) of a new random variable \( Y \), which likely relates to the given indicator function \( I_A(x) \) and the original continuous random variable \( X \) with its probability density function (PDF) \( f_X(x) \) and cumulative distribution function (CDF) \( F_X(x) \).

Let’s break it down step by step:

### Indicator Function

The indicator function \( I_A(x) \) for a set \( A \subseteq \mathbb{R} \) is defined as:

\[
I_A(x) =
\begin{cases}
1, & x \in A \\
0, & x \notin A
\end{cases}
\]

This function is used to determine whether a given \( x \) belongs to the set \( A \).

### Defining the Random Variable \( Y \)

Let \( Y = I_A(X) \), where \( X \) is a continuous random variable with PDF \( f_X(x) \) and CDF \( F_X(x) \). This implies that \( Y \) takes on two possible values:

- \( Y = 1 \) if \( X \in A \)
- \( Y = 0 \) if \( X \notin A \)

We want to find the CDF of \( Y \), denoted as \( F_Y(y) = \mathbb{P}(Y \leq y) \).

### Finding the CDF of \( Y \)

Since \( Y \) only takes the values 0 and 1, we can compute the probabilities for \( Y = 0 \) and \( Y = 1 \).

1. **For \( y < 0 \):**
   \[
   F_Y(y) = \mathbb{P}(Y \leq y) = 0
   \]
   because \( Y \) cannot take a value less than 0.

2. **For \( 0 \leq y < 1 \):**
   \[
   F_Y(y) = \mathbb{P}(Y \leq y) = \mathbb{P}(Y = 0)
   \]
   This is the probability that \( X \notin A \), which is:
   \[
   F_Y(y) = \mathbb{P}(X \notin A) = 1 - \mathbb{P}(X \in A)
   \]
   Therefore,
   \[
   F_Y(y) = 1 - \int_A f_X(x) \, dx
   \]
   where \( \int_A f_X(x) \, dx \) represents the probability that \( X \) lies within the set \( A \).

3. **For \( y \geq 1 \):**
   \[
   F_Y(y) = \mathbb{P}(Y \leq y) = 1
   \]
   because \( Y \) cannot take a value greater than 1.

### Final Expression for the CDF of \( Y \)

Thus, the CDF \( F_Y(y) \) is given by:

\[
F_Y(y) =
\begin{cases}
0, & y < 0 \\
1 - \int_A f_X(x) \, dx, & 0 \leq y < 1 \\
1, & y \geq 1
\end{cases}
\]

This expression describes the CDF of \( Y \) in terms of the indicator function and the PDF \( f_X(x) \) of the original random variable \( X \).

---

Would you like further details or explanations about this? Here are some related questions you might find interesting:

1. What is the relationship between a PDF and a CDF for a continuous random variable?
2. How do you compute the probability that a continuous random variable lies within a set \( A \)?
3. Can an indicator function be used for other transformations of random variables?
4. How would the CDF of \( Y \) change if \( A \) is an interval, like \( A = [a, b] \)?
5. How does the concept of measure theory relate to indicator functions?

**Tip**: Indicator functions are useful tools in probability for specifying when certain conditions hold, often simplifying computations with events and random variables.

Let X be a continuous random variable with pdf fX(x) and cdf FX(x). Let A be a subset of the real line. Let I_A(x) be the indicator function for A. Find an expression for the cdf of Y.

Find an expression for the CDF of a new random variable Y using the indicator function for a subset A, given a continuous random variable X with PDF fX(x) and CDF FX(x). This problem explores key concepts in probability theory and random variables, suitable for undergraduate students.

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