5.2.3 Example: Approximating a Standard Deviation

5.2.3  Example: Approximating a Standard Deviation

Suppose X ~ N(0,1), and consider the function


which is graphed in Exhibit 5.3.

Exhibit 5.3: Graph of the function f.

We use this function to define a new random variable Y = f(X). Although X is unbounded, we see in Exhibit 5.3 that Y is bounded, so the mean μ of Y must exist. We also see that f is symmetrical through the origin. Based upon this, and the symmetry of the standard normal distribution, we infer that the mean μ of Y is 0.

Suppose we need to determine the variance σ2 of Y. Because Y is bounded, this also must exist. Analytically solving for σ2 is a difficult problem, but we can easily approximate a solution with the Monte Carlo method. We define a sample {X[1], X[2], … , X[100]} for X, and construct a realization {x[1], x[2], … , x[100]} for this. We will discuss algorithms for doing so shortly. Our realization is indicated in Exhibit 5.4, and is plotted as a histogram in Exhibit 5.5.

Exhibit 5.4: A realization of a sample {X[1], X[2], … , X[100]} for X.
Exhibit 5.5: Histogram of the realization {x[1], x[2], …, x[199]}.

Based upon each value x[k], we use [5.7] to calculate the corresponding value y[k] = f(x[k]). Results comprise a realization {y[1], y[2], … , y[100]} of a sample {Y[1], Y[2], … , Y[100]} for Y and are indicated in Exhibits 5.6 and 5.7.

Exhibit 5.6: Values {y[1], y[2], … , y[100]} calculated from the x[k] of Exhibit 5.4 using [5.7].
Exhibit 5.7: Histogram of values y[k].

Because we know the mean of Y, we apply sample estimator [4.26] to the y[k] to obtain an approximate value for the variance σ2 of Y: