Consider the following function: ### Function DefinitionThe function $ f(x) $ is defined as:\[f(x) = \begin{cases} x/4 & \text{for } 0 \leq x < 1 \\1/8 & \text{for } 1 \leq x < 7 \\2 & \text{for } 7 \leq x < 113/16 \\0 & \text{else}\end{cases}\]### Tasks**(a)** Show that $ f(x) $ is a probability density function and sketch it.**(b)** Find the associated cumulative distribution function.**(c)** Let $ X $ be a random variable with probability density function $ f(x) $. From your sketch, decide whether it’s more likely that $ X $ will take a value in the interval $[1, 17/16]$ or $[7, 113/16]$.**(d)** Find the expectation of $ X $ and mark it on your sketch.### Graph DescriptionYou would typically sketch the function $ f(x) $ over the specified intervals. The graph should have a rising slope from zero at $ x=0 $ to $ 1/4 $ at $ x=1 $, a constant value of $ 1/8 $ from $ x=1 $ to $ x=7 $, and then a higher constant value of 2 from $ x=7 $ to $ x=113/16 $. Ensure the sketch highlights these distinct linear and constant pieces of the function over their respective intervals.

Answered: Image /qna-images/answer/d9a8864b-5b55-4a68-b4ca-4243ed1f4d49.jpg

Answered: f(x) = x/4 0

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Consider the following function:

### Function Definition

The function $ f(x) $ is defined as:

\[
f(x) =
\begin{cases}
x/4 & \text{for } 0 \leq x < 1 \\
1/8 & \text{for } 1 \leq x < 7 \\
2 & \text{for } 7 \leq x < 113/16 \\
0 & \text{else}
\end{cases}
\]

### Tasks

**(a)** Show that $ f(x) $ is a probability density function and sketch it.

**(b)** Find the associated cumulative distribution function.

**(c)** Let $ X $ be a random variable with probability density function $ f(x) $. From your sketch, decide whether it’s more likely that $ X $ will take a value in the interval $[1, 17/16]$ or $[7, 113/16]$.

**(d)** Find the expectation of $ X $ and mark it on your sketch.

### Graph Description

You would typically sketch the function $ f(x) $ over the specified intervals. The graph should have a rising slope from zero at $ x=0 $ to $ 1/4 $ at $ x=1 $, a constant value of $ 1/8 $ from $ x=1 $ to $ x=7 $, and then a higher constant value of 2 from $ x=7 $ to $ x=113/16 $. Ensure the sketch highlights these distinct linear and constant pieces of the function over their respective intervals.

Expert Solution

Step 1: Write the given information.

$f open parentheses x close parentheses equals open curly brackets table row cell x divided by 4 end cell cell 0 less or equal than x less than 1 end cell row cell 1 divided by 8 end cell cell 1 less or equal than x less than 7 end cell row 2 cell 7 less or equal than x less than 113 divided by 16 end cell row 0 cell text else end text end cell end table close$