Example 6.3. Compute the Monte Carlo estimate of the standard normal cumulative distribution function (cdf) [tex]\Phi(-1.96)[/tex] using an initial seed of 3700 and 10,000 iterations. Compare your estimate with the normal cdf function.

To compute the Monte Carlo estimate of the standard normal cdf for Φ(−1.96), generate 10,000 random numbers, count the amount that fall below -1.96, and then divide by 10,000. This estimate can be compared to the true normal cdf found using the standard formula, in line with the empirical rule.

Explanation:

To compute the Monte Carlo estimate of the standard normal cdf Φ(−1.96) using initial seed 3700 and 10000 iterations in a standard normal distribution where μ = 0 and σ = 1, you would generate 10,000 random numbers with the specified seed, count how many of these numbers fall below -1.96, and then divide that count by 10,000 to get the Monte Carlo estimate. You would then compare this with the true normal cdf using the formula normalcdf(lower value, upper value, mean, standard deviation), in this case, normalcdf(-1E99, -1.96, 0, 1).

The empirical normal distribution rule states that approximately 95 percent of the values are within two standard deviations of the mean, which provides the comparison to your Monte Carlo estimate.

This is a part of probability and statistics theory in mathematics where the use of the Monte Carlo method for estimation and the empirical rule are fundamental concepts.

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