💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here! # Let $f(x) = k(3x - x^2)$ if $0 \le x \le 3$ and $f(x) = 0$ if $x < 0$ or $x > 3$.(a) For what value of $k$ is $f$ a probability density function?(b) For that value of $k$, find $P (X > 1)$.(c) Find the mean.

## A. $k=\frac{2}{9}$B. $P(X > 1)=\frac{20}{27}$C. Mean $=\frac{27 k}{4}=\frac{3}{2}$

#### Topics

Applications of Integration

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##### Top Calculus 2 / BC Educators   ##### Samuel H.

University of Nottingham Lectures

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### Video Transcript

The question is there is a function given here like that. If excess between zero and three did a function takes this and otherwise function is there because on it said the request, the function is probably the best function. Pia By the definition off pita I can take the integral off PdF between a m. P. Excuse me, the proto X speed between and and when you replace age mind certainty that baby implemented then it gives me X is between mind 70 for angry and number that stood and is this one in that range is one and this is it goes toe one for NPR Is the future for Pete? Yes, So in the first part it asks for what is K So when you take is integral and take a difficult one and solve it I will replace ethics with our function. But our boundaries has changed with the pantry because when our X is less than zero, our ethics is because zero when our two e's X is greater that tree our function is zero. That means there is no effective integral X is not between geriatric. So this part must be equal to one then I will take that When I take the integral three. X becomes three X square over to and X car becomes X cubed over three and our boundaries Are you time skating? He goes to one. Then I just put three in tow instead of X. This part comes on when I could zero instead of X. This is cereal and there's a minus zero. I didn't write it, so I just calculate nine over to okay, because one that Kay he could still to over. Now this is the first book in the second part off questions it asks for What is a rowboat off X is great man. So I just used the first item here and replace a bit one baby infant When I replace 81 on TV Infinity This part becomes X's great development on poor built off X is greater than man Is Costa integral off ethics between one and infant. I just got here then when I replaced ethics with our function to over nine times three experts X square DX is integral and our boundaries are one tree like previous park. Then I just take integral three x become street external to and expert cups excuse over three on our families here. Then I put three into this part and it becomes one Then I could want instead of X. And this part comes then I just probably. And our boat off X is greater than oneness to end over plants. In the last part, it asks, or what is me off X When I just wantto find me, I just take the intractable X PdF Yes, in general. Then I just apply. When I put X fx, it becomes three x square and minus excuse Then I just take integral aunt to over nine times X coupe minus X for over four and our boundaries our country. And I just put three here on dhe. This part comes when I put zero here instead of X. It is It goes to zero, and I just calculate it is pre or an Army's 3/2 #### Topics

Applications of Integration

##### Top Calculus 2 / BC Educators   ##### Samuel H.

University of Nottingham Lectures

Join Bootcamp