Math Problem Statement
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.
Solution
To solve this problem, we need to find the cumulative distribution function (CDF) of a new random variable , which likely relates to the given indicator function and the original continuous random variable with its probability density function (PDF) and cumulative distribution function (CDF) .
Let’s break it down step by step:
Indicator Function
The indicator function for a set is defined as:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Indicator Functions
Continuous Random Variables
Cumulative Distribution Function (CDF)
Probability Density Function (PDF)
Formulas
I_A(x) = { 1 if x ∈ A, 0 if x ∉ A }
F_Y(y) = 0 for y < 0
F_Y(y) = 1 - ∫_A f_X(x) dx for 0 ≤ y < 1
F_Y(y) = 1 for y ≥ 1
Theorems
Properties of Cumulative Distribution Functions (CDF)
Properties of Indicator Functions
Suitable Grade Level
Undergraduate
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