91Ó°ÊÓ

A cumulative distribution function (CDF) is a fundamental concept in probability theory.
It describes the probability that a random variable takes on a value less than or equal to a given number.
For a continuous random variable \(X\) with a probability density function \(f(x)\), the CDF \(F(x)\) is defined as:
\[ F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) \ dt \] This means that \(F(x)\) accumulates the probabilities from the leftmost point of the distribution up to \(x\).
The CDF always ranges from 0 to 1, because the probability of \(X\) falling within its entire range is 1.
This cumulative property makes the CDF very useful in various applications, such as determining percentiles and constructing other distributions.

A probability density function (PDF) describes how the probabilities of a continuous random variable are distributed.
For a variable \(X\), the PDF is denoted by \(f(x)\).
The PDF has several key properties:

• The area under the curve \(f(x)\) over all possible values of \(X\) is 1.
• \(f(x)\) is always non-negative: \(f(x) \geq 0\) for all \(x\).
• The probability that \(X\) falls in an interval \([a, b]\) is the integral of \(f(x)\) from \(a\) to \(b\): \(P(a \leq X \leq b) = \int_{a}^{b} f(x) \ dx\).

Understanding the shape and properties of a PDF helps in interpreting the behavior of a random variable.
In the exercise, the PDF \(f(x)\) and its associated CDF \(F(x)\) are used to transform the random variable \(X\) into another variable \(Y\) to demonstrate uniform distribution.

An inverse function essentially reverses the operation of the original function.
For a given function \(f\), its inverse \(f^{-1}\) satisfies \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).
In probability theory, the inverse of the CDF \(F\) is crucial for transforming random variables.
When dealing with continuous random variables, the CDF \(F(x)\) is typically a monotonically increasing function, which means it is invertible.
Here's how the concept is used in the exercise:
To prove that the variable \(Y = F(X)\) is uniformly distributed between 0 and 1, we need to find its CDF \(G(y)\).
By expressing \(G(y)\) as \(P(Y \leq y)\), we identify that:
\(P(Y \leq y) = P(F(X) \leq y)\).
Since \(F\) is a CDF, it is invertible, hence:
\(P(F(X) \leq y) = P(X \leq F^{-1}(y))\).
Finally, substituting this into the definition of the CDF, we see that \(P(X \leq F^{-1}(y)) = y\).
Therefore, the CDF of \(Y\) is \(G(y) = y\), which confirms that \(Y\) is uniformly distributed.