Begin2.DVI

11-27. (Poisson distribution)

Let Xdenote a random variable representing the number of light bulbs which

fail during a specified time interval T. The random variable X is assumed to have

a Poisson probability density function where an average of 2 bulbs fail during the

time interval T. Use the Poisson probability density function

f(x) =^2

x

x!

e−^2 , x = 0, 1 , 2 ,...

and find the probability

(a) of no failures during time interval T

(b) of more than one failure during time interval T

(c) of more than five failures during time interval T

11-28. The error function or Gauss error function is defined^1

erf(x) = √^2

π

∫x

0

e−t

2

dt

(a) Show that

∫x

−∞

N(x; 0 ,1)dx =√^1

2 π

∫x

−∞

e−t

(^2) / 2
dt =^1
2
[

1 + erf

(

√x

2

)]

(b) Show that

1

σ

√

2 π

∫x

−∞

e−(

t−σμ)^2

dt =

1

2

[

1 + erf

(

x−μ

σ

√

2

)]

11-29. Each of the curves below represent graphs of the normal probability dis-

tribution N(x; 0 ,1). Explain how one would use areas associated with the normal

probability tables to find the areas of the shaded regions.

Find the area associated with the shaded areas.

(^1) There are alternative definitions for the error function.