homework questions. If preferable, you may type your answers. On any calculation

Question

homework questions. If preferable, you may type your answers. On any calculations, be sure to show all work to receive full credit and circle your final answer. 1, when examining a variable which has error, z-zo±or, the proba- bilities associated with samples of z are described mathematically by Prob(x) a e which is a gaussian function. The integrals of this function over varying values are what is displayed in the familiar t-score probability table. Now let's say we have two variables, a = ao ±oa and b-bo ±5b, and want to compute r- a+b. We know how to do this using our error propagation formulas, but we should also be able to get the same result using these probability distributions. This problem walks you through that calculation. (a) According to our error propagation formulas, what should be the average, To, and uncertainty, dr, of z a +b? With these, and the general form of the probability distribution shown above, what should be the probability distribution of z - ab? (b) Individually, a and b have probability distributions of the same form as the general one shown above. Together, their joint distri- bution is the product of their individuals, Prob(a,b) e- 2a -- as Modify the above equation by replacing a and b with: a+b = Note that the following mathematical identity will be quite useful. x2,y2 (x + y)2, (Ba-Ay) Page 1

Explanation / Answer

1. a. given two variables

a = ao +- da

b = bo +- db

hence from the error propogation formula

average

xo = (a + b)/2 = (ao + bo)/2

and

error in x = a + b is

dx = da + db

hence

form the given formula

Prob(x) = k*e^-(x - xo)^2/2dx^2

hence for x = a + b

Prob(x) = k*e^-((ao + bo)/2 +- (da + db))^2/2*(da + db)^2

2. given

Prob(a,b) = k(e^(-(a - ao)^2/2*da^2 - (b - bo)^2/2db^2))

let x = a + b

y = db^2(a - ao) - da^2(b - bo)

now,

a = x - b

y = db^2(x - b - ao) - da^2(b - bo)

hence

y = db^2*x - db^2*ao - db^2(b) - da^2(b) + da^2*bo

b = (db^2*x - db^2*ao + da^2*bo)/(db^2 + da^2)

b = (db^2*(x - ao) + da^2*bo)/(db^2 + da^2)

a = x - (db^2*(x - ao) + da^2*bo)/(db^2 + da^2) = (x db^2 + x*da^2 - db^2*(x - ao) - da^2*bo)/(db^2 + da^2)

a = (da^2(x - bo) - ao*db^2)/(db^2 + da^2)

hence

Prob(a,b) = k(e^(-((da^2(x - bo) - ao*db^2)/(db^2 + da^2) - ao)^2/2*da^2 - ((db^2*(x - ao) + da^2*bo)/(db^2 + da^2) - bo)^2/2db^2))

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