The CDF of X is given by F(x)= 1 for x ≥2 x^2 / 4 for 0 ≤x<2 0 for x<0 (a)

step 1 So we have this continuous random variable, and we want to, first of all, draw the graph of this

The CDF of X is given by F(x)= 1 for x ≥2 x^2 / 4 ...

The CDF of $X$ is given by $$ F(x)= \begin{cases}1 & \text { for } x \geq 2 \\ x^2 / 4 & \text { for } 0 \leq x<2 \\ 0 & \text { for } x<0\end{cases} $$ (a) find $f(x)$, the density function, and show that it satisfies the two requirements for a density function (b) graph $f(x)$ and $F(x)$.

The CDF of $X$ is given by
$$
F(x)= \begin{cases}1 & \text { for } x \geq 2 \\ x^2 / 4 & \text { for } 0 \leq x<2 \\ 0 & \text { for } x<0\end{cases}
$$
(a) find $f(x)$, the density function, and show that it satisfies the two requirements for a density function
(b) graph $f(x)$ and $F(x)$.

Key Concepts

Cumulative Distribution Function (CDF)

The CDF describes the probability that a random variable takes on a value less than or equal to a given number. It is a non-decreasing function that approaches 0 as the variable goes to negative infinity and 1 as it goes to positive infinity. In problems that involve determining densities or probabilities, the CDF serves as the starting point from which further descriptions such as the PDF can be derived.

Probability Density Function (PDF)

The PDF, applicable to continuous random variables, describes the relative likelihood for the variable to take on a specific value. It is obtained by differentiating the CDF with respect to the variable when the CDF is differentiable. A valid PDF must be non-negative over its domain and integrate to 1 over the entire space.

Piecewise-Defined Functions

Many functions in probability are defined piecewise, meaning different expressions are used over different intervals. This approach is particularly useful for modeling distributions that behave differently in separate regions of their support. When working with such functions, careful attention is required at the boundaries to ensure properties such as continuity of the CDF or proper integration of the PDF.

Differentiation of Functions

Differentiation is a key mathematical operation that connects the CDF to the PDF. For continuously differentiable segments of a CDF, the derivative provides the corresponding density function. In the context of piecewise functions, differentiation must be applied to each segment individually, while also checking the overall consistency of the resulting PDF.

Properties of Density Functions

A probability density function must always meet two essential criteria: it must be non-negative for all values within its domain, and the integral of the density over the entire space must equal 1. These properties ensure that the PDF accurately represents a probability distribution and that total probability is conserved.

Graphical Representation of Distributions

Graphing both the CDF and PDF provides a visual understanding of how probability is distributed across different values. The CDF graph shows the accumulation of probability, while the PDF graph illustrates the density or intensity of probability at each specific point. Visualization is a critical tool in probability and statistics for analyzing the behavior of distributions, especially when dealing with piecewise or non-standard functions.

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