Functions of a Random Variable

In summary, the problem involves finding the distribution of the random variable Y, which is defined as 1 if X is greater than 0 and -1 if X is less than or equal to 0. The equation P(Y <= y) = P(X belongs to g^(-1)(-inf, y]) is used to find the distribution. It is unclear if Y is a continuous or discrete random variable, and further clarification is needed to determine the appropriate steps to find the probability density function (PDF) for Y.
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ksm100
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Homework Statement


Let X be an RV of the continuous type, and let Y=g(X) be defined as g(x)=1 if x>0, and =-1 if x<=0. Find the distribution of Y.


Homework Equations


P(Y <= y) = P(X belongs to g^(-1)(-inf, y])


The Attempt at a Solution


I'm really not too sure what to do here so any help would be tremendously appreciated. For other questions where g is differentiable and the derivative is either <0 or >0 i know that Y will be a continuous RV and the expression for the PDF but here I'm confused.
 
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  • #2
Is Y a discrete RV? Do I just use the definition of Y to find the PDF? Any help would be greatly appreciated.
 

Related to Functions of a Random Variable

1. What is the definition of a random variable?

A random variable is a numerical value that is assigned to the outcomes of a random experiment. It represents the possible values that can result from the experiment.

2. What are the two types of random variables?

The two types of random variables are discrete and continuous. Discrete random variables can only take on a finite or countably infinite set of values, while continuous random variables can take on any value within a certain range.

3. What is the difference between a probability distribution and a probability density function?

A probability distribution is a mathematical function that describes the likelihood of each possible outcome of a random variable. A probability density function, on the other hand, is a type of probability distribution that is used for continuous random variables.

4. What are some real-world examples of random variables?

Some examples of random variables include the number of heads that will appear when flipping a coin, the temperature on a given day, and the time it takes to complete a task.

5. How are the expected value and variance of a random variable calculated?

The expected value of a random variable is calculated by multiplying each possible value by its corresponding probability and summing them all together. The variance is calculated by taking the sum of the squared differences between each possible value and the expected value, multiplied by their corresponding probabilities.

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