Download Chapter 3 Probability Distribution part 2

Chapter 3
Probability Distribution
Normal Distribution
Normal Distribution
• Numerous continuous variables have distribution closely
resemble the normal distribution.
• The normal distribution can be used to approximate
various discrete probability distribution.
A continuous random variable X is said to have a normal distribution
with parameters  and  2 , where       and  2  0,
if the pdf of X is
f ( x) 
1
 2
e
1  x 
 

2  
2
  x  
X is denoted by X ~ N (  ,  2 ) with E  X    and V  X    2
CHARACTERISTICS OF NORMAL
DISTRIBUTION
•‘Bell Shaped’
• Symmetrical
• Mean, Median and Mode
are Equal
Location is determined by the
mean, μ
Spread is determined by the
standard deviation, σ
The random variable has an infinite
theoretical range:
+  to  
f(X)
σ
X
μ
Mean = Median = Mode
Many Normal Distributions
By varying the parameters μ and σ, we obtain different
normal distributions
The Standard Normal Distribution

Any normal distribution (with any mean and standard deviation
combination) can be transformed into the standard normal
distribution (Z)

Need to transform X units into Z units using
X 
Z


The standardized normal distribution (Z) has a mean of ,   0
and a standard deviation of 1,  2  1

Z is denoted by Z ~ N (0,1)

Thus, its density function becomes
Calculating Probabilities for a General Normal
Random Variable
• Mostly, the probabilities involved x, a normal random
variable with mean, μ and standard deviation, σ
• Then, you have to standardized the interval of interest, writing it
in terms of z, the standard normal random variable.
• Once this is done, the probability of interest is the area that you
find using the standard normal probability distribution.
• Normal probability distribution, X~N ( μ, σ2 )
• Need to transform x to z using
Z
X 

Patterns for Finding Areas under the Standard Normal Curve
Example
Example
a) Find the area under the standard normal curve of
P (0  Z  1)
a) Find the area under the standard normal curve of
P(2.34  Z  0)
Exercise 3.1
Determine the probability or area for the portions of the
Normal distribution described.
a) P (0  Z  0.45)
b) P ( 2.02  Z  0)
c) P ( Z  0.87)
d) P ( 2.1  Z  3.11)
e) P (1.5  Z  2.55)
Answer : a) 0.1736, b) 0.4783, c) 0.8078, d) 0.9812,
e) 0.0614
Upper tail probabilities or areas of the distribution of
Z have also been tabulated. An entry in the table
specifies a value of Zα such that an area lies to its
right. In other word, P(Z>Z α)
Example 3.2
(a) P  Z  Z   0.16
(b) P  Z  Z   0.48
(c) P( Z  Z )  0.80
(d ) P  Z  Z   0.95
Exercise 3.2
Determine Z such that
a) P( Z  Z )  0.25
b) P( Z  Z )  0.36
c) P( Z  Z )  0.983
d) P( Z  Z )  0.89
Answers:
a) P( Z  Z )  0.25;
Z  0.6745
b) P( Z  Z )  0.36;
Z  0.3585
c) P( Z  Z )  0.983;
d) P( Z  Z )  0.89;
Z  2.1201
Z  1.2265
Example 3.3
Suppose X is a normal distribution N(25,25). Find
a) P(24  X  35)
b) P( X  20)
Z
Any normal distribution
can be transformed
into the standard
normal distribution
(Z)
X 

Example
Exercise 3.3:
1. A normal random variable x has mean, 10, and
standard deviation, 2. Find the probabilities below:
(a) P  X  13.5 (b) P  X  8.2  (c) P  9.4  X  10.6 
2. Hupper Corporation produces many types of soft drinks
including Orange Cola. It has been observed
that the net
amount of soda in such a can has a normal distribution with
a mean of 12 ounces and a standard deviation of 0.015
ounce. What Is the probability that a randomly selected can
of Orange Cola contains between 11.97 and 11.99 ounces of
soda?
3. The random variable X is normally distributed. Given
μ=54 and P ( X > 80 ) = 0.02. Find the value of σ.
Normal Approximation of the Binomial
Distribution
 When the number of observations or trials n in a binomial
experiment is relatively large, the normal probability
distribution can be used to approximate binomial
probabilities. A convenient rule is that such approximation is
acceptable when
n  30, and both np  5 and nq  5.
Given a random variable X b(n, p), if n  30 and both np  5
and nq  5, then X N (np, npq)
with   np and  2  npq
Continuous Correction Factor
The continuous correction factor needs to be made when a
continuous curve is being used to approximate discrete
probability distributions. 0.5 is added or subtracted as a
continuous correction factor according to the form of the
probability statement as follows:
c .c
a) P ( X  x) 
 P ( x  0.5  X  x  0.5)
c .c
b) P ( X  x) 
 P ( X  x  0.5)
c .c
c) P ( X  x) 
 P ( X  x  0.5)
c .c
d) P ( X  x) 
 P ( X  x  0.5)
c .c
e) P ( X  x) 
 P ( X  x  0.5)
c.c  continuous correction factor
How do calculate Binomial Probabilities
Using the Normal Approximation?
• Find the necessary values of n and p. Calculate μ = np and
  npq
• Write the probability you need in terms of X.
• Correct the value of x with appropriate continuous correction factor
(ccf).
• Convert the necessary x-values to z-values using
z
xccf  np
npq
• Use Standard Normal Table to calculate the approximate probability.
Example 3.5
In a certain country, 45% of registered
voters are male. If 300 registered
voters from that country are selected
at random, find the probability that at
least 155 are males.
Exercise 3.5
Suppose that 5% of the population over 70 years
old has disease A. Suppose a random sample of
9600 people over 70 is taken. What is the
probability that less than 500 of them have
disease A?
Answer: 0.8186
Normal Approximation of the Poisson
Distribution
 When the mean of a Poisson distribution is relatively
large,
the normal probability distribution can be
used to approximate Poisson probabilities. A
convenient rule is that such approximation is
acceptable when   10.
Given a random variable X
then X
N ( ,  )
Po ( ), if   10,

Example 3.6
A grocery store has an ATM machine inside. An
average of 5 customers per hour comes to use the
machine. What is the probability that more than
30 customers come to use the machine between
8.00 am and 5.00 pm?
Solution:
Exercise 3.6
The average number of accidental drowning in
United States per year is 3.0 per 100000
population. Find the probability that in a city of
population 400000 there will be less than 10
accidental drowning per year.
Answer : 0.2358
Extra
exercise
1)
A certain type of storage battery lasts, on average, 3.0 years with
a variance of 0.25 year. Assuming that the battery lives are
normally distributed, find the probability that a given battery will
last less than 2.3 years.
2)
An electrical firm manufactures light bulbs that have a life, before
burn-out, that is normally distributed with mean equal to 800
hours and a standard deviation of 40 hours. Find the probability
that a bulb burns between 778 and 834 hours.
3)
The probability that a patient recovers from a rare blood disease is
0.4. If 100 people are known to have contracted this disease, what
is the probability that less than 30 survive?
4)
Assume that the number of asbestos particles in a squared
meter of dust on a surface follows a Poisson distribution with a
mean of 1000. If a squared meter of dust is analyzed, what is
the probability that 950 or fewer particles are found?
5)
A process yields 10% defective items. If 100 items are
randomly selected from the process, what is the probability
that the number of defectives exceeds 13?