 # 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

[5.7]

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, X, … , X} for X, and construct a realization {x, x, … , x} 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, X, … , X} for X. Exhibit 5.5: Histogram of the realization {x, x, …, x}.

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, y, … , y} of a sample {Y, Y, … , Y} for Y and are indicated in Exhibits 5.6 and 5.7. Exhibit 5.6: Values {y, y, … , y} 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:

[5.8]