Cumulative distribution function

In summary, the conversation is about finding the cumulative distribution function of a new random variable, $Y=F_X (X)$, where $F_X (x)$ is a strictly monotone distribution function for the random variable $X$. The confusion arises from the use of the letter "x" in the notation, which can represent both a specific number and a random variable. By using the inverse function of $F_X$, $F_X^{-1}$, the cumulative distribution function of $Y$ can be simplified.
  • #1
Julio1
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Hi !, my problem is the following:

Let $F_X (x)$ an distribution function strictly monotone for the random variable $X$ and it's defined the new random variable $Y=F_X (X).$ Find the cumulative distribution function of $Y$.
In this case, $F_X (x)$ is an cdf, but I don't how does for find the cdf of $Y.$ I think that need have the density function, but I don't have is information.
 
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  • #2
Julio said:
Let $F_X (x)$ an distribution function strictly monotone for the random variable $X$ and it's defined the new random variable $Y=F_X (x).$
I don't understand the phrase "it's defined the new random variable $Y=F_X (x)$". $F_X(x)$, for each given number $x$, is a number. There is no random variable here. Recall that a random variable is a function from the sample space to $\Bbb R$. Do you mean $Y=F_X (X)$, i.e., $Y=F_X\circ X$?

In mathematics, uppercase and lowercase letters often denote different objects.
 
  • #3
Evgeny.Makarov said:
I don't understand the phrase "it's defined the new random variable $Y=F_X (x)$". $F_X(x)$, for each given number $x$, is a number. There is no random variable here. Recall that a random variable is a function from the sample space to $\Bbb R$. Do you mean $Y=F_X (X)$, i.e., $Y=F_X\circ X$?

In mathematics, uppercase and lowercase letters often denote different objects.

Thanks!, you're right, the correct is $Y=F_X (X).$ In this case, $Y=F_X(X)$ is an transformation?
 
  • #4
Julio said:
In this case, $Y=F_X(X)$ is an transformation?
I am not sure what you mean by this.

Since $F_X$ is strictly monotonic, it has an inverse $F_X^{-1}$. So
\[
F_Y(y)=\text{Pr}(Y<y)=\text{Pr}(F_X(X)<y)=\text{Pr}(F_X^{-1}(F_X(X))<F_X^{-1}(y))=\text{Pr}(X<F_X^{-1}(y))=\ldots
\]
Can you simplify this further?
 
  • #5


Hi there! The cumulative distribution function (CDF) of a random variable Y is defined as the probability that Y takes on a value less than or equal to a given value y. In this case, since Y is defined as a function of X, we can express the CDF of Y in terms of the CDF of X.

Using the definition of CDF, we can write:

$F_Y (y) = P(Y \leq y) = P(F_X (X) \leq y)$

Since F_X is strictly monotone, it is invertible and we can write:

$F_Y (y) = P(X \leq F_X^{-1} (y))$

Now, using the definition of CDF again, we can write:

$F_Y (y) = F_X (F_X^{-1} (y)) = y$

Therefore, the CDF of Y is simply y, which is a straight line with slope 1. This means that the probability of Y taking on a value less than or equal to y is equal to y itself.

I hope this helps! Let me know if you have any other questions.
 

FAQ: Cumulative distribution function

What is a cumulative distribution function (CDF)?

A cumulative distribution function (CDF) is a statistical function that maps the probability of a random variable being less than or equal to a certain value. It is used to describe the distribution of a continuous random variable.

How is a CDF different from a probability density function (PDF)?

A CDF describes the probability of a random variable being less than or equal to a certain value, while a PDF describes the relative likelihood of a random variable taking on a specific value. In other words, the CDF is the integral of the PDF.

What does the shape of a CDF tell us about the distribution of a random variable?

The shape of a CDF can tell us about the central tendency, variability, and skewness of a distribution. For example, a symmetrical CDF indicates a symmetric distribution, while a skewed CDF indicates a skewed distribution.

How can a CDF be useful in statistical analysis?

A CDF can be used to calculate probabilities for specific values or ranges of a continuous random variable. It can also be used to compare two or more distributions, or to generate random numbers from a specific distribution.

Can a CDF be used for discrete random variables?

Yes, a CDF can be used for both continuous and discrete random variables. For discrete random variables, the CDF is a step function, where the probability increases in steps at each possible value of the random variable.

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