Advanced High-School Mathematics

352 CHAPTER 6 Inferential Statistics

does the underlying density curve look like? Is it still uniform as in the

case ofX? Probably not, but let’s take a closer look. Before getting

√too heavily into the mathematics let’s start by collecting 200 samples of
randand drawing a histogram. This will give us a general idea of what
the underlying density curve might look like. Collecting the samples is
easy:
√
rand(200)→L 1

puts 200 samples from this distribution into the TI list variable L 1.

Likewise, this sampling is easily done using more advanced softwares

as Autograph or Minitab. Below is an Autograph-produced histogram

of these 200 samples.

We would suspect on the basis of this histogram that the underlying

density curve is not uniform but has considerable skew. Intuitively,

we could have seen this simply by noting that for any real numberx

satisfying 0< x <1 thenx <

√

x; this is what creates the histogram’s

to skew to the left.

Can we make this more precise? Yes, and it’s not too difficult. If we

setX=rand, Y =

√

rand, we have that for any value oft, 0 ≤t≤1,

P(Y ≤t) =P(

√

X≤t) =P(X≤t^2 ) =t^2

(sinceX is uniformly distributed on [0,1].) In other words, iffY is the

density function forY, then it follows that

∫t

0 fY(x)dx=P(Y ≤t) =t

(^2) ;