By generating 10 uniform random variate estimate the integral

Question

By generating 10 uniform random variate $U (0, 1)$ estimate the integral

$\theta = \frac{1}{\sqrt{2\pi}}\int_{-1}^{2}e^{\frac{-x^2}{2}} dx$

Recognizing this function as probability density function of $N (0, 1)$ , compare the value of $\widehat{\theta}$ with $\theta$ .

29 Feb 2024

Answer :

Word Count : 739

To estimate the integral using Monte Carlo simulation, we will generate 10 uniform random variates between 0 and 1, and then transform them to the range between -1 and 2 using the formula:

\[ x_i = -1 + (2 - (-1)) \times U_i \]

Then we will evaluate the function $ e^{\frac{-x^2}{2}} $ for each generated $ x_i $ and average them out.

Let's go through the steps:

1. Generate 10 uniform random variates, $ U_i $, between 0 and 1.
2. Transform each $ U_i $ to the range between -1 and 2 to get $ x_i $.
3. Evaluate the function $ e^{\frac{-x^2}{2}} $ for each $ x_i $.
4. Average the values obtained in step 3 to __________ ____ ______ ______ ____ ________ ____ _________ ____.
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