Mathematics, 07.03.2020 04:35 hunterbrod71
The time, in 100 hours, that a student uses her game console over a year is a random variable X with probability density function
f(x) = { x if 0 2-x if 1<=x<2
0 otherwise}
The power (in number of kilowatt hours) expended by the student's game console each year is 48X^2+26. For these problems, please ensure your answers are accurate to within 3 decimals.
Part a)
Find the mean amount of power expended by the student's game console per year.
Part b)
Find the variance of power expended by the student's game console per year.
Answers: 3
![\begin{array}{c}\\E\left( {{X^2}} \right) = \int\limits_0^1 {\left( {{x^2} \times x} \right)} \,\,dx + \int\limits_1^2 {{x^2}\left( {2 - x} \right)} \,\,dx\\\\ = \int\limits_0^1 {{x^3}} \,\,dx + \int\limits_1^2 {\left( {2{x^2} - {x^3}} \right)} \,\,dx\\\\ = \frac{1}{4}\left[ {{x^4}} \right] + \left[ {2\left[ {\frac{{{x^3}}}{3}} \right]_1^2 - \left[ {\frac{{{x^4}}}{4}} \right]_1^2} \right]\\\\ = \frac{1}{4} + \left[ {\frac{2}{3}\left( {{2^3} - {1^3}} \right) - \frac{1}{4}\left( {{2^4} - {1^4}} \right)} \right]=1.167\\\end{array}](/tpl/images/0537/7023/6e51f.png)
![\begin{array}{c}\\V\left( {48{X^2} + 26} \right) = {\left( {48} \right)^2}V\left( {{X^2}} \right)\\\\ = {48^2}\left( {E\left( {{X^4}} \right) - {{\left[ {E\left( {{X^2}} \right)} \right]}^2}} \right)\\\end{array}](/tpl/images/0537/7023/46510.png)
![\begin{array}{c}\\E\left( {{X^4}} \right) = \int\limits_0^1 {\left( {{x^4} \times x} \right)} \,\,dx + \int\limits_1^2 {{x^4}\left( {2 - x} \right)} \,\,dx\\\\ = \int\limits_0^1 {{x^4}} \,\,dx + \int\limits_1^2 {\left( {2{x^4} - {x^5}} \right)} \,\,dx\\\\ = \frac{1}{5}\left[ {{x^5}} \right]_0^1 + \left[ {2\left[ {\frac{{{x^5}}}{5}} \right]_1^2 - \left[ {\frac{{{x^6}}}{6}} \right]_1^2} \right]\\\\ = \frac{1}{5} + \left[ {\frac{2}{5}\left( {{2^5} - {1^5}} \right) - \frac{1}{6}\left( {{2^6} - {1^6}} \right)} \right]=2.1\\\end{array}](/tpl/images/0537/7023/dd02e.png)
![\begin{array}{c}\\V\left( {{X^2}} \right) = E\left( {{X^4}} \right) - {\left[ {E\left( {{X^2}} \right)} \right]^2}\\\\ = 2.1 - {\left( {1.167} \right)^2}\\\\ = 0.738\\\end{array}](/tpl/images/0537/7023/7cdf3.png)
Answer from: Quest
Just divide the x axis (the top) by 2 or multiply the y axis (the bottom) by 2
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