Below is the transcription of the image. There are no graphs or diagrams present:---Let $ X $ be the physical distance, measured in inches, from its anchoring point where a bicycle spoke will snap. Then $ X $ has the following probability density function: $ f(x) = \frac{6}{169}x\left(1 - \frac{x}{13}\right) $ for $ 0 \leq x \leq 13 $ and 0 otherwise.a) What is the probability that a bicycle spoke snaps less than 3 inches from its anchoring point? \[ 0.135 \]b) What is the probability that $ X > 6 $?\[ 0.558 \]c) What is the probability that $ 2 < X < 8 $?\[ 0.606 \]d) What is the expected value of $ X $ (E(X))?\[ 6.500 \]e) What is the expected value of $ X^2 $?\[ 50.696 \]f) What is the variance of $ X $?\[ 8.446 \]g) What is the standard deviation of $ X $?\[ 2.906 \]h) What is the probability that $ X $ is less than its expected value?\[ 0.500 \]i) What is the expected value of $ X^{0.6} $?\[ \]j) What is the 60th percentile of $ X $?\[ \]--- The answers to some questions are filled in, while some areas remain blank for user input.

Answered: Let X be physical distance, measured in…

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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Let $ X $ be the physical distance, measured in inches, from its anchoring point where a bicycle spoke will snap. Then $ X $ has the following probability density function: $ f(x) = \frac{6}{169}x\left(1 - \frac{x}{13}\right) $ for $ 0 \leq x \leq 13 $ and 0 otherwise.

a) What is the probability that a bicycle spoke snaps less than 3 inches from its anchoring point?
\[ 0.135 \]

b) What is the probability that $ X > 6 $?
\[ 0.558 \]

c) What is the probability that $ 2 < X < 8 $?
\[ 0.606 \]

d) What is the expected value of $ X $ (E(X))?
\[ 6.500 \]

e) What is the expected value of $ X^2 $?
\[ 50.696 \]

f) What is the variance of $ X $?
\[ 8.446 \]

g) What is the standard deviation of $ X $?
\[ 2.906 \]

h) What is the probability that $ X $ is less than its expected value?
\[ 0.500 \]

i) What is the expected value of $ X^{0.6} $?
\[ \]

j) What is the 60th percentile of $ X $?
\[ \]

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The answers to some questions are filled in, while some areas remain blank for user input.

Expert Solution

Introduction

[NOTE : As per the rules only the first 3 subparts are solved. ]

Given that,

$f (x) = \{\begin{cases} \frac{6}{169} x (1 - \frac{x}{13}) & 0 \leq x \leq 13 \\ 0 & o t h e r w i s e \end{cases}$

The following probabilities are to be obtained.

Step 2

(a) The probability that a bicycle spoke snaps is less than 3 is to be obtained.

Mathematically, It can be expressed as $\begin{array}{rcl} P \end{array} \begin{array}{rcl} (X < 3) \end{array}$ .

If $X$ denotes the bicycle spoke snaps then,

$\begin{array}{rcl} P (X < 3) & = & \int_{0}^{3} \frac{6}{169} x (1 - \frac{x}{13}) d x \\ = & \int_{0}^{3} \frac{6}{169} x d x - \int_{0}^{3} \frac{6}{2197} x^{2} d x \\ = & \frac{6}{169} {(\frac{x^{2}}{2})}_{0}^{3} - \frac{6}{2197} {(\frac{x^{3}}{3})}_{0}^{3} \\ = & \frac{6}{169} {(\frac{9}{2} - 0)}_{0}^{3} - \frac{6}{2197} {(\frac{27}{3} - 0)}_{0}^{3} \\ \approx & 0.1352 \end{array}$

Therefore the required probability that a bicycle spoke snaps is less than 3 inches from anchoring point is $0.1352$ .

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