ASSN11

course Mth 174

hysics II03-17-2008

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16:18:46

Query 8.8.2 (3d edition 8.7.2) 8.7.2. Probability and More On Distributions, p. 421 daily catch density function piecewise linear (2,.08) to (6.,24) to (8,.12)

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16:29:27

what is the mean daily catch?

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3.733 tons of fish / day

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16:32:30

What integral(s) did you perform to compute a mean daily catch?

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(2*.08) + (6*.24) + (8 *.12) / 3 = 3.8

since each day was only a fraction of a day it needed to be multiplyed by the time frame to account for it not being a full day.

This would be the first moment about the origin of a discrete function with values of .08, .24 and .12 units located respectively at x = 2, x = 6 and x = 8. The corresponding integral of a continuous function f(x) from x = a to x = b would be integral(x * f(x), x from a to b).

** You are asked here to find the mean value of a probability density function. From a probability density function you can only get a probability over an interval, and the result is obtained by integrating the function over that interval.

In this case the function is piecewise linear between the given points.

The linear functions that fit between the two points are y = .04 x and y = .6 - .06 x.

You should check to be sure that the integral of the probability density function is indeed 1, which is the case here.

The mean value of a distribution is the integral of x * p(x). In this case this gives us the integral of x * .04 x from x = 2 to x = 6, and x = x(.6 - .06 x) from x = 6 to x = 8.

int( x*.04 x, x, 2, 6) = .04 / 3 * (6^3-2^3) = .04*208/3=8.32/3 = 2.77 approx.

int(-.06x^2 + .6x, x, 6, 8) = [-.02 x^3 + .3 x^2 ] eval at limits = -.02 * (8^3 - 6^3) + .3 ( 8^2 - 6^2) = -.02 * 296 + .3 * 28= -5.92 + 8.4 = 2.48.

2.77 + 2.48 = 5.25.

The first moment of the probability function p(x) is the integral of x * p(x), which is identical to the integral used here. The mean value of a probability distribution is therefore its first moment. **

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16:34:32

What does this integral have to do with the moment integrals calculated in Section 8.3?

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It is the the same as calculating density because it is a density function and we are trying to calculate the average density. Where as before we were calculating the mass.

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16:42:29

Query 8.8.13 (3d edition 8.7.13). Probability and More On Distributions, p. 423 cos t, 0

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16:45:04

which function might best represent the probability for the time the next customer walks in?

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B- because as t moves away from 0 the probability desity goes down that someone else will enter at that second.

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16:53:33

for each of the given functions, explain why it is either appropriate or inappropriate to the situation?

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A- this doesn't have a smooth change, it is the same then jumps up with the expnential function.

B- Is corect, it represents the bell shape of distribution to what were looking for.

C- This function is close but is very steep comparitivly to the B function.

D- this is a constant and wouldn't work because we're looking for a changing function.

** Our function must be a probabiity density function, which is the case for most but not all of the functions.

It must also fit the situation.

If we choose the 1/4 function then the probability of the next customer walking in sometime during the next 4 minutes is 100%. That's not the case--it might be 5 or 10 minutes before the next customer shows up. Nothing can guarantee a customer in the next 4 minutes.

The cosine function fluctuates between positive and negative, decreasing and increasing. A probability density function cannot be negative, which eliminates this choice.

This leaves us with the choice between the two exponential functions.

If we integrate e^(-3t) from t = 0 to t = 4 we get -1/3 e^-12 - (-1/3 e^0 ) = 1/3 (1 - e^-12), which is slightly less than 1/3. This integral from 0 to infinity will in fact converge to 1/3, not to 1 as a probability distribution must do.

We have therefore eliminated three of the possibilities.

If we integrate 3 e^(-3t) from 0 to 4 we get 1 - e^-12, which is almost 1. This makes the function a probability density function. Furthermore it is a decreasing function. **

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16:54:40

Query Add comments on any surprises or insights you experienced as a result of this assignment.

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Applying the integral to some real things is much easier to understand than just the theories.

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See my notes and be sure you understand everything. Let me know if you have questions.