Begin2.DVI

11-30. If Φ(z) = √^1

2 π

∫z

−∞

e−ξ

(^2) / 2

dξ, find the value of and illustrate with sketches

the representations of the following probabilities as shaded area under the normal

probability density curve.

(a) P(Z≤1)

(b) P(Z > 1)

(c) P(Z≤ 3 .2)

(d) P(Z≤− 3 .2)

(e) P(− 1. 2 ≤Z≤ 0 .75)

(f) P(|Z|≤ z)

(g) P(Z≤2)

(h) P(|Z|≤ 2

(i) P(|Z|≤ 3)

11-31. (Monte Carlo computer problem )

(a) Give a physical interpretation to the integral I 1 =

1

b−a

∫b

a

f(x)dx

(b) Give a physical interpretation to the summation I 2 =

1

N

∑N

i=1

f(xi)where

a≤xi≤bfor all integers i.

(c) If I 1 =I 2 , show estimate for I=

∫b

a

f(x)dx is given by I=

b−a

N

∑N

i=1

f(xi)

(d) Calculate 500 random numbers^2 xiwith xi∈(− 1 ,2) and estimate the integral

I=√^12 π

∫ 2

− 1 e

−x^2 / (^2) dx. Do this over and over again and calculate 1000 estimates

for I and then do descriptive statistics on your results and compare your

computer answer with the answer obtained from table lookup.

11-32. (Monte Carlo computer problem)

Write a Monte Carlo computer program to calculate the area bounded by

the curves y=e−x

(^2) / 2

and y= 0. 4 as illustrated below. Set up as an integral (see

previous problem) or throw darts at area.

Hint 1: y=e−x

(^2) / 2

= 0. 4 when x=x∗=±

√

−2 ln(0.4)

Hint 2: Construct a rectangle where −x∗ < x < x ∗

and 0. 4 ≤y ≤ 1. 0 about area to be calculated by

Monte Carlo method.

Hint 3: Generate random numbers (xr, yr) with

−x∗ ≤xr ≤x∗ and 0. 4 ≤yr ≤ 1. 0 and determine if

the point (xr, y r)is inside or outside the area to be

determined.

Be sure to perform descriptive statistics on your

results and if your instructor gives you extra credit,

put confidence intervals on your answer for the area.