Question

in part A. We are going to find the value of Okay, so we know that the integral awful. The pdf, It should be one and in this case the integral is from 0 to 1. This is okay X cubed dx. So it is K over four X to the fourth. 0 1 plug in and subtract. So we get que over for here. So this implies that K equal four in part B. We are going to compute the variance of X. The various of ads. It is the expectation of X square minus that. The square of the expectation of X. So we need to compute the expectation of X which is Integral from 0 to 1 X F x of X dx. This is Integral from 0 to 1 four X To the four DX. So this is 4/5 X to the five, 01 plugging and subtract, This is four or 5. And we still need to compute the expectation of X square Which is integral from 0 to 1, X square times P D F of X dx. This is integral from 0 to 1 four X to the power of five DX. So it is 4/6 X to the power six plug in one rule and subtract. So we get For over six which is 2/3. So the various of x, it is 2/3 minus the square awful. 4/5. So it is, we take common denominator which is 75. And here we put two times 25, 50 minus three times um 16 which is 48. So this is two over 75. Part C. We are going to find the value of M. So such as that the probability X less than or equal to M. Yes, It's equal to 1/2. So the probability X less than or equal to M. This is the integral from 0 to M. F X. Of X. D X. So this is Integral from 0 to M. four x cube D X. This is X to the 4th 02 M. And subtract. So it is M to the power for So we have M E kuo the false route offer one half. Finally for part D in party we are we define this why equal to X plus two And we are going to find the pdf of y. So we know X. Take the value On the Interval 01. This implies that why takes value the interval, We plug in zero and plug in one. We get two and four respectively be. So why takes value on 24? So now we compute this probability why less than equal to little one. This is the probability two X plus two less than equal to y. Which is the probability that X is less than equal to one half Y minus what? So this is the integral From a 0- one half. Why minus one F X. Of X. D X. Here we have computed this this integral of the p D F of X from zero to M. M. M. To the power four. So it is, One have Y -1 to the power for. So F Y. Y. It is the derivative of this with respect to Y. So it is so this So be four, one half, y minus one to the power three, and then the derivative of this interior things. So it is one half. So it is equal to two. one have Y -1 Cube, or To less than what? Less than four.