91Ó°ÊÓ

The cumulative distribution function (CDF) is a fundamental tool in probability and statistics used to describe the probability that a random variable takes on a value less than or equal to a certain threshold. For a continuous random variable, the CDF is the integral of the probability density function (PDF), which in this case we denote as \(f_X(x)\). The CDF is expressed mathematically as: \[ F_X(x) = \int_{-\infty}^{x} f_X(t) \, dt \] When dealing with a specific interval for a variable \(x\), only the section where the PDF is non-zero is considered. In our example, \(f_X(x)\) is non-zero from 0 to 2, so the CDF is calculated over this range. The CDF provides the total probability accumulated up to a given point \(x\) in the interval. This helps in understanding the distribution and likelihood of different outcomes for the variable. Importantly, the CDF always increases with \(x\) since probabilities cannot decrease, and it approaches 1 as \(x\) approaches infinity, covering the entire probability space.

The probability density function (PDF) is crucial in defining the likelihood of a continuous random variable. Unlike the discrete counterpart, for which probabilities of exact values are defined, a PDF gives a density over an interval and thus can yield probabilities only in the context of an interval. For a random variable \(X\) with PDF \(f_X(x)\), the function must satisfy the property that:

• \(f_X(x) \geq 0\) for all \(x\)
• \(\int_{-\infty}^{\infty} f_X(x) \, dx = 1\)

These conditions ensure that the total probability sums to 1 and no negative probabilities are assigned. Understanding a PDF allows us to determine CDF by integration. From the CDF, one can also derive back to the PDF by differentiation. For the given function, the specific form describes how probabilities distribute over the interval 0 to 2. Notably, outside this interval, the PDF is naturally addressed by being zero, implying no probability beyond these bounds.

Transforming variables is a common technique used to study how one variable relates to another via a functional relationship. When a function of a random variable is involved, the distribution of this new variable can be derived through transformation rules. For a random variable \(X\) and a transformation \(Y = g(X)\), the new random variable's distribution needs to be evaluated. Critical steps include using the inverse transformation and correctly adjusting for changes in probability intervals. When we transform \(X\) to \(Y = \sqrt{X}\), observing the range of \(Y\) is essential, which in this context changes from 0 to \(\sqrt{2}\). To find the CDF of \(Y\), we look at probabilities up to a certain \(y\), expressed as: \[ F_Y(y) = P(Y \leq y) = P(\sqrt{X} \leq y) = P(X \leq y^2) = F_X(y^2) \] The PDF for \(Y\) can subsequently be found by differentiating the CDF. These steps are foundational in analyzing how transformations affect data and help reveal insights that may not be evident in the original variable's distribution.