Math Problem Statement
Solution
Let's solve the problem step by step:
(a) Determine the cumulative density function of for all :
The given probability density function (PDF) is:
\begin{cases} 2x^3 + x, & 0 \leq x \leq 1, \\ 0, & \text{otherwise}. \end{cases}$$ The cumulative density function (CDF) $$F(x)$$ is the integral of $$f(x)$$ from $$-\infty$$ to $$x$$: $$F(x) = \int_{-\infty}^x f(t) \, dt.$$ 1. For $$x < 0$$: $$F(x) = 0$$, since $$f(x) = 0$$ for $$x < 0$$. 2. For $$0 \leq x \leq 1$$: $$F(x) = \int_0^x (2t^3 + t) \, dt.$$ Perform the integration: $$\int (2t^3 + t) \, dt = \frac{2t^4}{4} + \frac{t^2}{2} = \frac{t^4}{2} + \frac{t^2}{2}.$$ Thus: $$F(x) = \left[\frac{t^4}{2} + \frac{t^2}{2}\right]_0^x = \frac{x^4}{2} + \frac{x^2}{2}.$$ 3. For $$x > 1$$: $$F(x) = 1$$, since the total probability sums to 1. Final result: $$F(x) = \begin{cases} 0, & x < 0, \\ \frac{x^4}{2} + \frac{x^2}{2}, & 0 \leq x \leq 1, \\ 1, & x > 1. \end{cases}$$ --- ### **(b) Construct an algorithm for generating random variates for $$X$$ using the composition method:** To generate random variates for $$X$$, we utilize the inverse transform sampling method. The steps are as follows: 1. Generate a uniform random number $$U \sim U(0, 1)$$. 2. Set $$U = F(X)$$, where $$F(X) = \frac{X^4}{2} + \frac{X^2}{2}$$ for $$0 \leq X \leq 1$$. 3. Solve for $$X$$ in terms of $$U$$: $$U = \frac{X^4}{2} + \frac{X^2}{2}.$$ Let $$v = 2U$$, so: $$v = X^4 + X^2.$$ This is a nonlinear equation that can be solved numerically. 4. Use a numerical root-finding algorithm (e.g., Newton-Raphson) to solve for $$X$$ in $$0 \leq X \leq 1$$. --- ### **(c) Linear Congruential Generator (LCG):** The LCG is given as: $$Z_i = (7Z_{i-1} + 2) \mod 16, \quad Z_0 = 6.$$ The uniform random numbers are: $$U_i = \frac{Z_i}{16}.$$ #### Compute $$Z_i$$ and $$U_i$$ for enough $$i$$ to cover one full cycle: Start with $$Z_0 = 6$$ and iterate the recurrence relation: 1. $$Z_1 = (7 \cdot 6 + 2) \mod 16 = 44 \mod 16 = 12$$. $$U_1 = \frac{12}{16} = 0.75.$$ 2. $$Z_2 = (7 \cdot 12 + 2) \mod 16 = 86 \mod 16 = 6$$. $$U_2 = \frac{6}{16} = 0.375.$$ The sequence $$Z_i$$ repeats after 2 steps: $$Z_0 = 6$$, $$Z_1 = 12$$, $$Z_2 = 6$$. Therefore, the period $$p$$ is 2. #### First $$X$$ generated: The first random number $$U_1 = 0.75$$ corresponds to solving: $$F(X) = 0.75.$$ Solve $$\frac{X^4}{2} + \frac{X^2}{2} = 0.75$$ numerically. --- Would you like a detailed breakdown of any part or further explanation? Here are some related questions to deepen understanding: 1. How is the CDF derived for more complex probability density functions? 2. Can you explain how numerical root-finding algorithms like Newton-Raphson work? 3. What are the advantages of using the composition method in generating random variates? 4. How can the period of an LCG impact the quality of random numbers? 5. What are common methods to validate the uniformity of a random number generator? **Tip:** When solving nonlinear equations numerically, always check the range of possible solutions to ensure convergence!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability density function (PDF)
Cumulative density function (CDF)
Numerical integration
Random variate generation
Linear congruential generator (LCG)
Formulas
F(x) = integral from -∞ to x of f(t) dt
Z_i = (aZ_(i-1) + c) mod m
U_i = Z_i / m
Numerical solution of equations (root-finding algorithms)
Theorems
Fundamental theorem of calculus
Inverse transform sampling theorem
Suitable Grade Level
Undergraduate/Advanced High School
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