Transcript Example

PROBABILITY AND
STATISTICS
WEEK 7
Onur Doğan
Normal Distributions
Onur Doğan
Normal Probability Distributions
• The normal probability distribution is the most
important distribution in all of statistics
• Many continuous random variables have normal
or approximately normal distributions
• Need to learn how to describe a normal
probability distribution
Normal Distributions
Onur Doğan
Normal Distributions
Onur Doğan
Standardization
Standart Normal Distribution
The standard normal random variable (denoted
as Z) is a normal random variable with mean µ=
0 and variance Var(X) = 1.
Onur Doğan
Standard Normal Distribution
Properties:
• The total area under the normal curve is equal to 1
• The distribution is mounded and symmetric; it extends
indefinitely in both directions, approaching but never
touching the horizontal axis
• The distribution has a mean of 0 and a standard deviation
of 1
• The mean divides the area in half, 0.50 on each side
• Nearly all the area is between z = -3.00 and z = 3.00
Standardization
Standart Normal Distributions
Onur Doğan
Example
 Example: Find the area under the standard normal curve between
z = 0 and z = 1.45
0
z
0.00
0.01
0.02
0.03
145
.
0.04
z
0.05
..
.
1.4
..
.
P (0  z  145
. )  0.4265
0.4265
0.06
Example (Reading the Z Table)
• P(0 ≤ Z ≤ 1,24) =
• P(-1,5 ≤ Z ≤ 0) =
• P(Z > 0,35)=
• P(Z ≤ 2,15)=
• P(0,73 ≤ Z ≤ 1,64)=
• P(-0,5 ≤ Z ≤ 0,75) =
• Find a value of Z, say, z , such that P(Z ≤ z)=0,99
Onur Doğan
Example
Onur Doğan
Example
• A debitor pays back his debt with the avarage
45 days and variance is 100 days. Find the
probability of a person’s paying back his debt;
a) Between 43 and 47 days
b) Less then 42 days.
c) More then 49 days.
Onur Doğan
Example
The sick-leave time of employees in a firm in a
month is normally distributed with a mean of
100 hours and a standard deviation of 20 hours.
Find the probability that the sick-leave time of an
employee in a month exceeds 130 hours.
Onur Doğan
Onur Doğan
Approximation to Normal Distribution
Onur Doğan
Normal Approximation to the Binomial
Distributions
n=20 and p=0.6
Onur Doğan
Normal Approximation to the Binomial
Distributions
The binomial distribution B(n,p) approximates to
the normal distribution with;
E(X)= np and Var(X)= np(1 - p) if
np > 5 and n(l -p) > 5
Onur Doğan
Example
Suppose that X is a binomial random variable
with n = 100 and p = 0.1.
Find the probability P(X≤15) based on the
corresponding binomial distribution and
approximate normal distribution. Is the normal
approximation reasonable?
Onur Doğan
Normal Approximation to the Poisson
Distributions
The normal approximation is applicable to a
Poisson if λ > 5
Accordingly, when normal approximation is
applicable, the probability of a Poisson random
variable X with µ=λ and Var(X)= λ can be
determined by using the standard normal random
variable
Onur Doğan
Example
• Suppose that X has a Poisson distribution with
λ= 10. Find the probability P(X≤15) based on
the corresponding Poisson distribution and
approximate normal distribution. Is the normal
approximation reasonable?
Onur Doğan
Normal Approximation to the
Hypergeometric Distributions
Recall that the binomial approximation is
applicable to a hypergeometric if the sample size
n is relatively small to the population size N, i.e.,
to n/N < 0.1.
Consequently, the normal approximation can be
applied to the hypergeometric distribution with
p =K/N (K: number of successes in N) if
n/N < 0.1, np > 5. and n(1 - p) > 5.
Onur Doğan
Example
Suppose that X has a hypergeometric distribution
with N = 1,000, K = 100, and n = 100. Find the
probability P(X≤15) based on the corresponding
hypergeometric distribution and approximate
normal distribution. Is the normal approximation
reasonable? (δ=2,85)
Onur Doğan
Examples
Onur Doğan
Example
For a product daily avarege sales are 36 and
standard deviation is 9. (The sales have normal
distribution)
a) Whats the probability of the sales will be less
then 12 for a day?
b) The probability of non carrying cost
(stoksuzluk maliyeti) to be maximum 10%,
How many products should be stocked?
Onur Doğan
Example
Onur Doğan
Example
Onur Doğan
Example
Onur Doğan