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The completion time $X$ for a task has cdf $F(x)$ given by$$\left\{\begin{array}{cc}{0} & {x<0} \\ {\frac{x^{3}}{3}} & {0 \leq x<1} \\ {1-\frac{1}{2}\left(\frac{7}{3}-x\right)\left(\frac{7}{4}-\frac{3}{4} x\right)} & {1 \leq x \leq \frac{7}{3}} \\ {1} & {x \geq \frac{7}{3}}\end{array}\right.$$(a) Obtain the pdf $f(x)$ and sketch its graph.(b) Compute $P(.5 \leq X \leq 2) .$(c) Compute $E(X)$

Intro Stats / AP Statistics

Chapter 3

Continuous Random Variables and Probability Distributions

Section 9

Supplementary Exercises

Continuous Random Variables

University of North Carolina at Chapel Hill

Piedmont College

Cairn University

Boston College

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So in this particular question, they have given us the CDF. Ah, so, uh, I want to simplify. And the first part that they have told us so the CDF is given in the question waas. So first part is to find out the media for this FX and also brought the laugh. So you know that pdf is given by the derivative off the CDF with respect works. So in this case, now, the values that we have been given because on that if I apply the derivatives so zero will remain like that on me and ah, David Wolfe. Uh, X cube by three will be three x square by three. So in cans and three. And this is for zero less unequal to excellence and one and then the next part of the limited Wanna zero. So we have minor, soft as it is, and the remaining protons are buying you. Really? So what I'd get here is, say, seven upon three minus x into derivative off their bracket. Part is minus three by four. Bless, then, uh, and keeping seven upon four, minus three by four X as it is. And here the dead of it off. This is minus one. And this is for values from one tau seven by tree. And the next part is one for X value greater than equal to seven by three. So if I have you and this whole part by simplifying the expression, I'm finally getting the PdF smaller for fixed as this will be zero as it is for the first part, then the next, But is X squared for zero less than equal to X less than one. And then this part after simplification is one upon four into seven minus three x. This is from one less than equal to X, less than equal to seven by three. And the last part is again one. Uh, yeah, I think we forgot it in to date. A minute of your So this negative. Why here will change to zero because they never do off constantly. Zeros mismo changed a zero. And here also this zero for X is greater than equal to seven by three. So based on this effort, friends that we have, you're supposed to knock around the graph for this so we won't get really value on the X and buy access. So if I talk about the skill now, you okay? Go x on the x axis. But if you observe the values for the XX is the scale that we will be taking. Bree, you see, uh, it is ah, Indian from, say, Well, 0 to 1 and then 1 to 7.3. So I'm actually going to take it directly, say, from minus going five Okay, up to three. Because of a values between these values and for the y axis, which is actually going to be a four by 4 30 if off x that we really plotting. So for that, if you substitute the values off except we have a washer will get the values off for fix and that we can find out that it is going to be between zero and one, cause pdf should know always positive. So we take the value will here as the older one. So the graph waiter that you need, I have already done that. So I'm going to take them. Graph yours? Yeah. So this is the graph. Then we have more. Don't just thatyou size and get into the proper form here. Yeah, so we can now observe the graph. Getty. It is shown that initially, uh or here you can see that initially, the values are stable AB zero for the values off eggs. But later on, it is rising. And again, after a certain weir of time are reaching the peak value, it doesn't a decreasing more, and it reaches 20 war hero. So this is the graph for this particular PdF Now moving further. They have also told us to find out uh, the program beauties. So for the meat bark, they told us to find the probably off ex lying between Quien five. And so, if you take the definition of the probability, this will range from 0.52 and that ever do off ffx dx. But well, here are enough effects. If you see the range of quiet fighter too, we will have to consider two functions your X squared as the less this part because the force functions from 0 to 1 and it's from one through seven country. So simplifying this part by spreading the indignation over Sure, uh, I want to simplify it in this way so we can therefore see the scandal din as integral from 0.5 to 1. Ffx DX missed the first part and further from one to to the next partners if affects the X as artists. So substituting the values that I have here from 0.5 to 1 if affects is actually X Square and then from 1 to 2, the value off FX is given as one by four into seven, minus three X gx. So far, blaming dilation. This is excused by three from 0.51 plus one by four. As it is, this will be seven x minus three. Into this is X square by two indignation off X and this is from one toe. So substituting the limits, the first part will be simplified using cans You can take it at seven upon 24. Plus The next part is subsumed the upper limit minus a little immature. Want to simplify it using the cancer to get last five by eight. So I think what the fractions descends. It is 11 a born drill and you can finally like that as your 0.916 not moving for that. They have also told to find the expectation for this particular PdF So the formula for expectation. Off X is given by integral minus infinity to infinity X and Y f affects the X. So our popular some is ranging from 0 to 7 by three some eight this into forms again from 0 to 1. We have the function X into f x x squared DX and plus from 1 to 7 upon three. The function is X into one upon four seven minus three x d x. So, applying the indication here we have X cube So the indignation be expressed to four by four from 01 And for this part, if you multiply X in the black a cure and then integrate what we're going to get here is one by four, as it is in the bracket the Thomas seven X So the integration of the seven X square by two minus three X squares and really three X cube by three and the whole limit is wrong. One toe, seven of one tree. So apple immaterial people. This will be won by four okay, less. After putting the upper limit minus and women over here, we simplify the expressions. What we're going to get is 26 by 27. So the finance it is 1 31 of 100 eat. And if you simplify, I'm there for getting the value of the mean over here as 1.21

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