If a pdf of Y is ƒ(y)={1-|>| ||S1' />1 find and graph F(y).
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![### Problem Statement
If a probability density function (pdf) of \( Y \) is
\[
f(y) =
\begin{cases}
0, & |y| > 1 \\
1 - |y|, & |y| \leq 1
\end{cases}
\]
find and graph \( F(y) \).
### Explanation
**Probability Density Function (pdf):**
- The function \( f(y) \) describes the probability density function for the random variable \( Y \).
- For \(|y| > 1\), the pdf is zero, indicating that values of \( Y \) outside this range have a probability density of zero.
- For \(|y| \leq 1\), the pdf is given by \(1 - |y|\), a linear function that forms a triangle with base endpoints at \( y = -1 \) and \( y = 1 \), peaking at \( y = 0 \).
**Cumulative Distribution Function (CDF):**
The cumulative distribution function \( F(y) \) is obtained by integrating the pdf:
- For \( y < -1 \), \( F(y) = 0 \).
- For \(-1 \leq y \leq 0\), calculate the integral from \(-1\) to \( y \) of \(1 - (-t)\).
\[ F(y) = \int_{-1}^{y} (1 - (-t)) \, dt = \int_{-1}^{y} (1 + t) \, dt \]
Calculate the integral to find \( F(y) \) in this interval.
- For \( 0 < y \leq 1\), calculate the integral from \(-1\) to \( 0\) and then add the integral from \(0\) to \( y \) of \(1 - t\).
\[ F(y) = \int_{-1}^{0} (1 + t) \, dt + \int_{0}^{y} (1 - t) \, dt \]
Continue calculating to find the complete CDF \( F(y) \).
- For \( y > 1 \), \( F(y) = 1 \), as all probabilities sum to 1.
**Graph:**
- Graph \( F(y) \), which will be a continuous, non-decreasing function.
- The graph of the pdf](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67df2f2b-9148-4dd5-b0ae-87328374df62%2Fde28f381-f90c-40c4-8dd4-a5b04280eda3%2Fzqx2xag_processed.jpeg&w=3840&q=75)
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