Chapter 10: Q. 10.2 (page 431)

Give an approach for simulating a random variable having probability density function
f(x) = 30(x2 − 2x3 + x4) 0 < x < 1

Short Answer

Expert verified

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

Step by step solution

Given Information

We have given the density function

$f (x) = 30 (x^{2} - 2 x^{3} + x^{4}) 0 < x < 1$

Simplify

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

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

Now, using the differentiation, we have

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

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

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

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

$U \leq \frac{f (Y)}{c g (Y)}$

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

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Give an approach for simulating a random variable having probability density function
f(x) = 30(x2 − 2x3 + x4) 0 < x < 1

Short Answer

Step by step solution

Given Information

Simplify

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Give an approach for simulating a random variable having probability density functionf(x) = 30(x2 − 2x3 + x4) 0 < x < 1

Short Answer

Step by step solution

Given Information

Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Calculus

Probability and Statistics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Give an approach for simulating a random variable having probability density function
f(x) = 30(x2 − 2x3 + x4) 0 < x < 1