Prob. Rando var and density function
- Thread starter galbax02
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Please show us what you have tried and exactly where you are stuck.
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Please share your work/thoughts about this problem.
pka
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Note that \(\displaystyle\int_0^1 {30({x^2} - 2{x^3} + {x^4})dx}=1 \)
How would you this to build a pdf?
Following my other post we note that in this case [MATH]F(x)[/MATH] is bijective over it's support and thus we don't have to do anything fancy.
However....
[MATH]F(x) = 6 x^5-15 x^4+10 x^3[/MATH][MATH]F^{-1}(x)[/MATH] has no closed form.
What it appears you'll have to do is generate the Uniform[0,1] variate [MATH]u[/MATH] as before
and then solve for [MATH]F(x)=u,~0<x<1[/MATH], and then that solution is your random variate with the pdf given by [MATH]f(x)[/MATH]
Alternatively......
You could make use of the fact that [MATH]f(x)[/MATH] looks an awful lot like a truncated normal distribution.
You could find the parameters that are best fit for the given polynomial and apply well known methods for generating normal variates.
If some fall outside the support region (0,1) they are discarded.
However....
[MATH]F(x) = 6 x^5-15 x^4+10 x^3[/MATH][MATH]F^{-1}(x)[/MATH] has no closed form.
What it appears you'll have to do is generate the Uniform[0,1] variate [MATH]u[/MATH] as before
and then solve for [MATH]F(x)=u,~0<x<1[/MATH], and then that solution is your random variate with the pdf given by [MATH]f(x)[/MATH]
Alternatively......
You could make use of the fact that [MATH]f(x)[/MATH] looks an awful lot like a truncated normal distribution.
You could find the parameters that are best fit for the given polynomial and apply well known methods for generating normal variates.
If some fall outside the support region (0,1) they are discarded.