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A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.

(a) Explain why the function

$$ f(x) = \left\{

\begin{array}{ll}

0.1 & \mbox{if $ 0 \le x \le 10 $}\\

0 & \mbox{if $ x < 0 $ or $ x > 10 $ }

\end{array} \right.$$

is a probability density function for the spinner's values.

(b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.

A. $=[0.1 x]_{0}^{10}=0.1(10)=1$

B. Mean $=5$

Applications of Integration

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Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

we have. Ah, spinner. That birthdates gives a number between between zero and then So we have here older or spinner when you put in your state. So it is, uh, all this level here is gonna be then and then those from zero so immediately close with the zero. So one. So she's going to give up the range of numbers between zero. So the outcome there's gonna be any number any riel number between, uh, zero under one. I'm 10. So it's Ah, it's fair. So that means that Oh, uh, old numbers have the same. Well, the numbers between you got into behalf, same product, same probability this probability has a distribution. Yeah, if you fix that is, um good one if X is, uh, between Syria and 10 Macedo zero. Otherwise, um, so this is, uh, is a probability density because, well, uh, you know, cases It is, uh, be here. Down or zero. This is the first conditional. You need to check on that. Well, secondly, Misha, check that, uh, the reading between, like, as a lot of possibilities. Okay, so this is the set of all possibilities. You integrate your bags, the eggs you should be one. Yes, The probability off the whole space should be one was a s is gonna be, well, the real numbers. But since this is cereal out off the intervals, zero up to 10, this in the world is gonna be dangerous. All off this number, Uh, from zero up them. Well, that is a constant, so we can pull it out of the integral. On the interval off the X is just the X this evaluated between off that times to appoint one between zero and then which is time then? My no zero times 0.1 that is equal to what So we'll lease satisfies this condition. That is also true. So checks the first condition of the second condition. So it is. It's a probability density on the well, since, uh, it is, uh it is fair. The the average should be average move should be equal to or expire. However one wants to call, it should be. The middle should be the midpoint between zero. I'm 10. Which world? The midpoint is five. So Well, that's what the intuition says. But what we can compute what is discouraged because he's gonna be ableto the X with terms the way you affects the X. Also, this would be able to the integral from zero up to 10 effects your effects, the eggs, and then all that is 0.1 so we can pull out that constant out of the intervals. It's gonna be 2.1 times going to go from zero off, 10 off. Thanks. The X on. Didn't you seem to go all this integral is, uh, mhm. Sexist word. She's X squared excess. Perhaps you can 0 10 on that damn syrup 0.1. So this is going toe to your 0.1. Thanks. Thumbs are then squared hubs. Uh, so Super one time stand is one, so it's gonna be equal toe one times 10 ups on that issue. Goto. Right. So this is the average? Uh, yeah, British is. They're not everything. Syrian energies. Five. So bangs the average

University of Colorado at Boulder

Applications of Integration