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6> 0. I know you can d [ Select ] e kernel trick. 1/b ution. Many distributions can be s) times the kernel (the terms that Step 1: We first need to broken into a normalizing a cannot be separated fron e^(-x/b) Here, you recognize that [Select ] is the kernel of an exponential distribution f(x) = de-A, and the support of an exponential distribution is (0, oo) --- that is, the limits of integration here. Step 2: Use the distribution to figure out what the normalizing constant would be. In the exponential distribution, X is the normalizing constant. In the kernel you identified above,1 = [ Select ] [ Select] 1/b divide by the normalizing constant, and obtain the integrand You rearrange the terms in your integral so that it has the form e^(-x/b) c f(x) dæ where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form [ Select ] roper exponential distribution and c is some constant. ab (1/b) * e^(-x/b) Choose the other answer. It's right. Step 4: Since f(x) is the pdf of an exponential distribution, o f(x)dx = 1. Thus, ae /b dx = (your expression for c) %3D Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form c f(x)dx where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] [ Select ] a/b Step 4: Si a*b ential distribution, f(x)dx = 1. Thus, S ae Step 4: Since f(x) is the pdf of an exponential distribution, So f(x)da = 1. Thus, ae a/b dx = (your expression for c) %3D

6> 0. I know you can d [ Select ] e kernel trick. 1/b ution. Many distributions can be s) times the kernel (the terms that Step 1: We first need to broken into a normalizing a cannot be separated fron e^(-x/b) Here, you recognize that [Select ] is the kernel of an exponential distribution f(x) = de-A, and the support of an exponential distribution is (0, oo) --- that is, the limits of integration here. Step 2: Use the distribution to figure out what the normalizing constant would be. In the exponential distribution, X is the normalizing constant. In the kernel you identified above,1 = [ Select ] [ Select] 1/b divide by the normalizing constant, and obtain the integrand You rearrange the terms in your integral so that it has the form e^(-x/b) c f(x) dæ where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form [ Select ] roper exponential distribution and c is some constant. ab (1/b) * e^(-x/b) Choose the other answer. It's right. Step 4: Since f(x) is the pdf of an exponential distribution, o f(x)dx = 1. Thus, ae /b dx = (your expression for c) %3D Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form c f(x)dx where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] [ Select ] a/b Step 4: Si a*b ential distribution, f(x)dx = 1. Thus, S ae Step 4: Since f(x) is the pdf of an exponential distribution, So f(x)da = 1. Thus, ae a/b dx = (your expression for c) %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION: (use the dropdowns to select the correct answers) We wish to do the following integral: ае x/ dx, where a and b are some constants, with 6 > 0. I know you can do this by hand, but try it with the kernel trick. Step 1: We first need to recognize the kernel of a distribution. Many distributions can be broken into a normalizing constant (involving parameters) times the kernel (the terms that cannot be separated from x).
Transcribed Image Text:The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION: (use the dropdowns to select the correct answers) We wish to do the following integral: ае x/ dx, where a and b are some constants, with 6 > 0. I know you can do this by hand, but try it with the kernel trick. Step 1: We first need to recognize the kernel of a distribution. Many distributions can be broken into a normalizing constant (involving parameters) times the kernel (the terms that cannot be separated from x).
6> 0. I know you can d [ Select ] e kernel trick. 1/b ution. Many distributions can be s) times the kernel (the terms that Step 1: We first need to broken into a normalizing a cannot be separated fron e^(-x/b) Here, you recognize that [ Select ] is the kernel of an exponential distribution f(x) = de-A, and the support of an exponential distribution is (0, o) --- that is, the limits of integration here. Step 2: Use the distribution to figure out what the normalizing constant would be. In the exponential distribution, X is the normalizing constant. In the kernel you identified above,1 = [ Select ] [ Select ] 1/b divide by the normalizing constant, and obtain the integrand You rearrange the terms in your integral so that it has the form e^(-x/b) c f(x)dx where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form [ Select ] roper exponential distribution and c is some constant. ab (1/b) * e^(-x/b) Choose the other answer. It's right. Step 4: Since f(x) is the pdf of an exponential distribution, o f(x)dx = 1. Thus, ae a/b dx = (your expression for e) %3D Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form c f(x)dx where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] [ Select ] a/b Step 4: Si a*b ential distribution, f(x)dx = 1. Thus, S ae Step 4: Since f(x) is the pdf of an exponential distribution, So f(x)da = 1. Thus, S ae-a/b da = (your expression for c)
Transcribed Image Text:6> 0. I know you can d [ Select ] e kernel trick. 1/b ution. Many distributions can be s) times the kernel (the terms that Step 1: We first need to broken into a normalizing a cannot be separated fron e^(-x/b) Here, you recognize that [ Select ] is the kernel of an exponential distribution f(x) = de-A, and the support of an exponential distribution is (0, o) --- that is, the limits of integration here. Step 2: Use the distribution to figure out what the normalizing constant would be. In the exponential distribution, X is the normalizing constant. In the kernel you identified above,1 = [ Select ] [ Select ] 1/b divide by the normalizing constant, and obtain the integrand You rearrange the terms in your integral so that it has the form e^(-x/b) c f(x)dx where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form [ Select ] roper exponential distribution and c is some constant. ab (1/b) * e^(-x/b) Choose the other answer. It's right. Step 4: Since f(x) is the pdf of an exponential distribution, o f(x)dx = 1. Thus, ae a/b dx = (your expression for e) %3D Step 3: Thus, you multiply and divide by the normalizing constant, and obtain the integrand [ Select ] You rearrange the terms in your integral so that it has the form c f(x)dx where f(x) is a proper exponential distribution and c is some constant. Then c = [ Select ] [ Select ] a/b Step 4: Si a*b ential distribution, f(x)dx = 1. Thus, S ae Step 4: Since f(x) is the pdf of an exponential distribution, So f(x)da = 1. Thus, S ae-a/b da = (your expression for c)
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