SECTION 6.2 Continuous Random Variables 351
In this case thenormalrandom variableXcan assume any real value.
Furthermore, it is true—but not entirely trivial to show by elementary
means—that
√^1
2 πσ∫∞
−∞
e−(^12) (x−σμ)^2
dx= 1,
which by now we should recognize as being a basic property of any
density function.
Example. Our graphing calculators allow for sampling from normal
distributions, via therandNorm(μ,σ,n), where nis the number of in-
dependent samples taken. The calculator operation
randNorm(1, 2 ,200)→L 1
amounts to selecting 200 samples from a normally-distributed popula-
tion havingμ= 1 andσ= 2. The same can be done in Autograph; the
results of such a sample are indicated below:
6.2.2 Densities and simulations
In the above we had quite a bit to say about density functions and about
sampling from the uniform and normal distributions. We’ll continue
this theme here.
Let’s begin with the following question. Suppose thatXis the uni-
form random number generator on our TI calculators:X=rand. Let’s
define a new random variable by setting Y =
√
X =
√
rand. What