(a) Explain why the function whose graph is shown is a probability density function.
(b) Use the graph to find the following probabilities:
(i) $ P (X < 3) $ (ii) $ P (3 \le X \le 8) $
(c) Calculate the mean.
A. therefore $f(x)$ is a probability density function.
B. $P(X<3)=0.15$ and $P(3 \leq X \leq 8)=0.75$
Applications of Integration
for part A. We first want to prove that this is a probability density function. And so we want the area under the curve over its entire domain to some toe one. And so this is actually just the area of a triangle on. We could do one half base, which is 10 times a tight, which is 0.2, but 0.2 is really 2/10. So let's just change that now. Too tense. And then, of course, the 10 cancels and the two cancels. And so we really are just left with an area of one. And so this is continuous positive, and it's area sums toe one over its domain. So this is a probability density function for part B. We want to find the Probability X is less than three. And so we just want to find the area under the curve to the left here and looking at the, um, triangle there, we're gonna have one half the base, which is three times the height, which is 0.1. And so just writing this is a fraction that would be 1/10 and so multiplying straight up across that be 3/20 and 3/20 is the same thing as 0.15 or 15%. So find the area or the probability that an event happens between three and X. We want to some the area under the curve from there. So let's do that in black. So we want to find the area from here to here. Basically, there's gonna be a rectangle that we could add. And so let's go ahead and do that rectangle first, and then we'll add the triangle on top of it. And so that rectangle there has a base of five, has a height of 0.1, and so we're gonna move by, folks, Sorry, just five times There a 0.1 because it's just length times with or we could say, five times 1/10. But five tents is the same thing as one half, so that rectangle has an area of one half. But then we want to add the triangle, which is on top of it, since that also goes from three 28 and that would conclude the area under the curve there off that top section. So that triangle straw that over here also has a base of five. So it be one. Yeah, face of five. And it also has a height of 0.1. And so the 0.1, of course, is just 1/10 and so multiplying. Actually, we could just simplify, because then five goes into 10 twice and then we get up top one times 112 times 2/4. And so both of these then added together the triangle, plus the rectangle below it. One half plus 1/4 gives us this probability here ends up adding to three for its or 0.75 Okay, so we're just geometrically splitting up the area here, and then the last part is a bit tricky because we want to calculate the main. So we're going to find the equation of the line here of the graph and that has goes from 0 to 6. We multiply that function by X, so it's X times ffx. But f of X is just, um, the rise over the rum. Is there a point to over six and then tons X. So again, this is the line that then we multiply it by X. Since the formula is thus to calculate the main. So I hope that makes sense. And then the X all of that then plus the area under the curve from the rest of it, which is from six for 10. But then this section over here, we use red again. There we go, that section. If we continued it up, it would have a Y intercept of 0.5. And so that's going to be alright. X times this function strong red. Well, the slope is negative and it's negative. We go down 0.2 over four. So is your point to over four X plus 0.5 The Y intercept. All of that added together will give us the mean and then the mean just double check is going to come out to 16 3rd. So it's right that blue. So the mean both of these in a girl's added together gives us 16 3rd