Advanced High-School Mathematics

SECTION 6.2 Continuous Random Variables 355

(c) the probability that the quadraticx^2 +Bx+C = 0 has two

real roots.

4. Do the same as above where you take 200 samples from a normal
   distribution having μ = 0 and σ = 1. Create a histogram and
   draw the corresponding normal density curve simultaneously on
   your TI calculators.^14


5. Define the random variable by settingZ=rand^2.

(a) Determine the density function forZ. Before you start, why

do you expect the density curve to be skewed to theright?

(b) Collect 200 samples of Z and draw the corresponding his-

togram.

(c) How well does your histogram in (b) conform with the density

function you derived in (a)?

6. Consider the density functiongdefined by setting

g(t) =







4 t+ 2 if −^12 ≤t≤ 0

− 4 t+ 2 if 0 ≤t≤^12

6

A

A

A

A

A

A

A

AA

1/2

t

y=g(t)

(a) Show thatY =^12 X 1 +^12 X 2 −^12 , whereX 1 =rand,X 2 =rand,

X 1 andX 2 are independent. (Hint: just draw a picture in the

X 1 X 2 -plane to computeP(a≤Y ≤b).)

(^14) In drawing your histogram, you will need to make note of the widths of the histogram bars in
order to get a good match between the histogram and the normal density curve. For example, if you
use histogram bars each of width .5, then with 200 samples the total area under the histogram will
be. 5 ×200 = 100. Therefore, in superimposing the normal density curve you’ll need to multiply by
100 to get total area of 100. (UseY 1 = 100∗normalpdf(X, 0 ,1).)