Let X denote the amount of space occupied by an article placed in a 1-ft packing container. The pdf of X is below. 90x (1 - - x) 0 < x < 1 f(x) = otherwise (a) Graph the pdf. f(x) f(x)
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![The image appears to show an assignment response form with several probability-related questions and inputs. Here's a transcription suitable for an educational website:
---
### Assignment Questions:
**(b) What is \( P(X \leq 0.6) \) [i.e., \( F(0.6) \)]?**
*Round your answer to four decimal places.*
- Answer: **0.0464** ✔️
**(c) Using the cdf from (a), what is \( P(0.3 < X \leq 0.6) \)?**
*Round your answer to four decimal places.*
- Answer: **0.0462** ✔️
**(d) What is the 75th percentile of the distribution?**
*Round your answer to four decimal places.*
- Answer: **0.9036** ✔️
**(e) Compute \( E(X) \) and \( \sigma_X \).**
*Round your answers to four decimal places.*
- \( E(X) = \) **0.0123** ✖️
- \( \sigma_X = \) **0.1114** ✔️
**(f) What is the probability that \( X \) is more than 1 standard deviation from its mean value?**
*Round your answer to four decimal places.*
- Answer: **0.3136** ✖️
---
For clarification:
- Each question is listed with a designated letter.
- Correct answers are marked with a check (✔️), while incorrect answers have a cross (✖️).
- There is an option for additional help: "Need Help? Read It" button included.
No graphs or diagrams are included in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F431ffcf4-5182-4538-b673-402f10341839%2F02cfc360-4826-4720-a5be-2396a4485d63%2Fimzjfdh_processed.jpeg&w=3840&q=75)
![**Probability Density Function of Space Occupied by an Article**
Let \( X \) denote the amount of space occupied by an article placed in a \(1\text{-ft}^3\) packing container. The probability density function (pdf) of \( X \) is given by:
\[
f(x) =
\begin{cases}
90x^8(1-x) & \text{if } 0 < x < 1, \\
0 & \text{otherwise.}
\end{cases}
\]
### (a) Graph the pdf
**Graph Description:**
1. **Left Graph:**
- **Axes:** The horizontal axis represents \( x \), ranging from 0 to 1. The vertical axis represents \( f(x) \) with values ranging approximately from 0 to 4.
- **Curve:** The graph of \( f(x) \) starts at (0, 0) and follows a trajectory that initially rises steeply before reaching a peak around \( x = 0.8 \). It then descends sharply as it approaches \( x = 1 \).
2. **Right Graph:**
- **Axes:** Similar to the left graph, the horizontal axis indicates \( x \) (0 to 1), and the vertical axis represents \( f(x) \) (0 to 4).
- **Curve:** This graph has a similar shape, illustrating the same function with a peak near \( x = 0.8 \), resembling part of the Beta distribution's behavior.
### Summary
These graphs depict the probability density function of the occupied space \( X \) in a packing container. The function indicates that the likelihood of occupying a specific amount diminishes as it approaches the container's full capacity or emptiness, peaking around a middle value. This behavior aligns with a bell-shaped curve that illustrates a typical usage pattern in such contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F431ffcf4-5182-4538-b673-402f10341839%2F02cfc360-4826-4720-a5be-2396a4485d63%2Fa1k442e_processed.jpeg&w=3840&q=75)
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