Question

Give an approach for simulating a random variable having probability density function

f(x) = 30(x2 − 2x3 + x4) 0 < x < 1

Short answer

The method for simulating a random variable is the rejection method.

Step-by-step solution

Given Information

We have given the density function

$f\left(x\right)=30(x^2-2x^3+x^4)0<x<1$

Simplify

The PDF of $Y$is $g\left(y\right)=1$for $0<y<1.$Using the rejection method. Let's determine the value of $c.$We have

$\frac{f\left(x\right)}{g\left(y\right)}=f\left(y\right)=30(y^2-2y^3+y^4)$

Now, using the differentiation, we have

$\frac{df}{dy}=30(2y-6y^2+4y^3)=60y(1-3y+2y^2)=0$

Solving this equation, we have $\frac{df}{dy}=0$if and only if $y=0,1,1/2.$So, we can write the upper bound

$30(y^2-2y^3+y^4)\le f\left(\frac{1}{2}\right)=\frac{15}{8}$

Therefore, we can take $c=\frac{15}{8}.$Now, the method is given as. Generate $Y$and $u$uniformly from $(0,1)$and independent and consider whether

$U\le \frac{f\left(Y\right)}{cg\left(Y\right)}$

If it's true, declare $X=Y$. Otherwise, repeat these steps.