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Let $ f(x) = 30x^2 (1 - x)^2 $ for $ 0 \le x \le 1 $ and $ f(x) = 0 $ for all other values of $ x $.(a) Verify that $ f $ is a probability density function.(b) Find $ P(X \le \frac{1}{3}) $.

(A) $$\left[10 x^{3}-15 x^{4}+6 x^{5}\right]_{0}^{1}=10-15+6=1$$(B) $$\frac{10}{27}-\frac{15}{81}+\frac{6}{243}=\frac{17}{81}$$

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 5

Probability

Applications of Integration

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all right, so we're given this function. I felt backs and into the first thing. We like to show that after Maxie's their proper imitated function, which means we need to show to thes the 1st 1 is to show that half of axe is unknown. Actual function, which makes Apple back Please go to your quarters your forks. It's like a second that you have to show that to the integration off our backs. Backs for a minus Infinity Jin Tae's what? Or the 1st 1 history forward? Because the only thing we have to worry about it's this axe fortitude and to what minus X squared. But they're square functions, so both of them are good. Record was yours, which means 30 times accessible two times one minds acts tore up to It's good for 1/4 Duke four. That's and this simply impart that hour after backs. They're not negative reaction ever backs is quite what what is your forks? And secondly, we have to show that this intrusion is one. But if you look at this function again, the only numbers he repeats of this factory's 30 times extra, two times one minus extra, too, and They say that with access between Jonah What? So this integration is actually so cities into pushing for a minor symphony to finish your apple ax? The ax in our case is just to the introduction from Georgia. One 30. Acts are two types. What? Mine is excellent. Yes. Now they see the inaugural piece and this up run over limit so that it depends on the interval of these acts. All right, so that's, you know, I do it to this introduction. So because 30 skater, we can take 30 outside. I wonder what I'm going to do now is to expand the bracket 30 times integration from one extra two terms, one minus two acts purse act for opportunity axe and then this is actual to minus to execute Plus, actually fourth. And then by the general rule off intuition. This year, 30 times 1/3 of Axe Cube to my studio work for extra four plus one or five after the fifth on record Tree batter this for middle toe. What so perky in the upper limit? Have 1/3 time's one minus one over shoot this one world trade. Just this chill or four times one puzzling over five times away on minus in Europe because we're probably in the lower limit is your ring in the formula we got here and you will see that this year just 30 times toe one or 30. So this is what are so full party we have to find what's the probably Tina acts smaller or you quit your third. So again, by definition, this is just the intuition from 0 to 1/3 half of ax the ax. And again, the only numbers were a piece of life of axes when access between zero and one. So this is something the Twitter from 0 to 1/3 13 axe about 21 mind sacks to the about you? Yes, and from one we just found out with the intuition. These this is 30 times the third act's cute minus one or two acts to the fools, plus 1/5 ax to her five. But in this time we're country vitamin. This function from this is your actual third so again puppy in the upper limit. I have the third time's the third cubed minus half times 1/3 to 4 plus of faith times Ah, third toe fight on minus a picture here because we're puck in the lower limb is you're in the formula again because you and if you use your calculator awesome computer software, you'll see that this is so please 17 over anyway.

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