Find the value of a **Text Description:**The random variable $ x $ has the following continuous probability distribution in the range $ 0 < x < a $, as shown in the figure below.**Graph Explanation:**The graph is a two-dimensional plot showing a continuous probability distribution function $ f(x) $ over the interval $ 0 < x < a $.- The horizontal axis, denoted as $ x $, represents the range of the random variable, extending from $ 0 $ to $ a $.- The vertical axis, labeled $ f(x) $, represents the probability density function value at each point $ x $.The function $ f(x) $ is shown as a straight line sloping downward from the point $(0, a)$ on the vertical axis to the point $(a, 0)$ on the horizontal axis, forming a right triangle. This line indicates that the probability density decreases linearly from its maximum value $ a $ at $ x = 0 $ to zero at $ x = a $.

Name: Find the value of a **Text Description:**The random variable $ x $ h
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Description: Find the value of a **Text Description:**The random variable $ x $ has the following continuous probability distribution in the range $ 0 < x < a $, as shown in the figure below.**Graph Explanation:**The graph is a two-dimensional plot showing a

Answered: Find the value of a

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Find the value of a

**Text Description:**

The random variable $ x $ has the following continuous probability distribution in the range $ 0 < x < a $, as shown in the figure below.

**Graph Explanation:**

The graph is a two-dimensional plot showing a continuous probability distribution function $ f(x) $ over the interval $ 0 < x < a $.

- The horizontal axis, denoted as $ x $, represents the range of the random variable, extending from $ 0 $ to $ a $.
- The vertical axis, labeled $ f(x) $, represents the probability density function value at each point $ x $.

The function $ f(x) $ is shown as a straight line sloping downward from the point $(0, a)$ on the vertical axis to the point $(a, 0)$ on the horizontal axis, forming a right triangle. This line indicates that the probability density decreases linearly from its maximum value $ a $ at $ x = 0 $ to zero at $ x = a $.

Expert Solution

Step 1

A function $f : ℝ \to [0, 1]$ is said to be a probability density function if

i. $f (x) \geq 0$ and

ii. $\int_{- \infty}^{\infty} f (x) d x = 1$ .

Step 2

The given figure shows that as x increases f(x) decreases.

Hence,

$f (x) = \{\begin{cases} (a - x), & if 0 < x < a \\ 0 & otherwise . \end{cases}$

$Now, \int_{0}^{a} (a - x) d x = 1 \Rightarrow \int_{0}^{a} (x - a) d x = - 1 \Rightarrow {\frac{{(x - a)}^{2}}{2}}_{0}^{a} = - 1 \Rightarrow 0 - \frac{a^{2}}{2} = - 1 \Rightarrow a^{2} = 2 \Rightarrow a = \sqrt{2}, since a > 0 .$

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