$Finding two importance functions$

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Throwaway3

Matchmaticians Finding two importance functions File #1

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Kav10

Low bounty. I can solve this for you, if you increase the bounty to at least $20. Thanks.
Aman R

Do you want it via software method ?
- Throwaway3
  
  No. I just need the two importance functions. No need for any workings out.
Mathe

Bounty is too low, I'm afraid.

Kav10

Low bounty. I can solve this for you, if you increase the bounty to at least $20. Thanks.
Aman R

Do you want it via software method ?

Throwaway3

No. I just need the two importance functions. No need for any workings out.
Throwaway3

No. I just need the two importance functions. No need for any workings out.

Answer 1

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Here are the two importance functions:

f1(x) = exp(-x) x?(2,?)
f2(x) = x*e^{(1-x^2)/4}/sqrt(2*pi))) x?(2,?)

In case you want to see the functions graphs, here is an R code:

x <- seq(1,20,0.05)
y <- 1/sqrt(2*pi)* exp((-x^2/2))

y1 <- exp(-x)
y2 <- 1 / (x^2)

gs <- c(expression(g(x)==e^{-x^2/2}/sqrt(2*pi)),expression(f[1](x)==1/(x^2)),expression(f[2](x)==x*e^{(1-x^2)/4}/sqrt(2*pi)))

par(mfrow=c(1,2))

plot(x, y, type = "l", ylab = "", ylim = c(0,0.5),main='density function')
lines(x, y1, lty = 2,col="blue")
lines(x, y2, lty = 3,col="orange")
legend("topright", legend = 1:3, lty = 1:3,inset = 0.04,col=c("black","blue","orange"))

Please note that the bounty was too low for this question. A tip is appreciated. Thanks.

Kav10

Throwaway3

Hi Kav, thank you for the response, it's much appreciated. I sent you a small tip, apologies for it not being a lot.
- Kav10
  
  Thanks a lot! I am glad you liked the response.

$Finding two importance functions$

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