Download Normal Distribution - Engineering Perspectives - Lecture Slides and more Slides Process Engineering in PDF only on Docsity!
Normal Distribution
- The normal distribution (also known as Gaussian distribution)
with parameters μ and σ , N ( μ ,σ), is the continuous probability
distribution with the following probability density function:
- Normal distribution is the cornerstone of the field of
statistical inference
- Many distributions can be well approximated by the normal
distribution
- e.g. weight, height, bolt diameter, resistance of wire, construction error, etc.
1
− −
⋅
f x e x
x
X
2 2
( )^2
2
(^1) σ
μ
σ π
Where π = 3.14159…, e = 2.71828…
(Eq. 1)
Normal Distribution
2
Graph of the Probability Dense
Function of Normal Distribution
- A bell curve, symmetric about X = μ
- Shape of the curve is determined by σ
- the larger σ is, the flatter the curve is
σ 2 > σ 1
4
μ
Z =
Z
P( Z <1) = 0.
What is the total area under curve?
5
P( Z <- z ) = P ( Z > z ), for z >
Z =-1^ Z =
μ
P( Z <-1) = 0.159 (^) P( Z >1) = 0.
Z
7
Suppose the resistance of a carbon composition resistor is normally
distributed with mean μ = 1,000 and variance σ^2 = 900. What is the
probability that the resistor has measurement in excess of 1,060 ohms?
X ~ N ( μ =1000, σ=30), P ( X >1060)=?
ZX = ( X -1000)/
P ( X >1060)= P ( ZX > (1060-1000)/30) = P ( ZX >2) = 1- P ( ZX < 2)
X = the resistance of a carbon composition
Define X
Student’s t-Distribution
- Student’s t-distribution or t-distribution is used to replace normal
distribution when the standard deviation σ is not known and thus
has to be estimated using sample standard deviation s
8
df = ∞ (becomes normal distribution) df = 10 df = 5 df = 2 df = 1
Confidence Interval
- A confidence interval gives an estimated range of values
which is likely to include an unknown population parameter;
it is calculated from a given set of sample data
- Confidence Level
- How sure the confidence interval includes the true value of the
population parameter to be estimated
10
“We are 95% confident that between 60% and 70% people will agree
with this proposal.”
95% is our confidence level; (60%, 70%) is our confidence interval
Confidence Interval for Mean
- Suppose a random sample of size n is drawn from a normal
distribution with mean μ and standard deviation σ
(unknown), then the confidence interval for μ is:
11
( ( 1 ) , ( 1 ) )^ (Eq. 6)
/ 2 / (^2) n
s
n
s
x − t n − ⋅ x + t n − ⋅
α α
where α is significance level and equal to (1 – confidence level)
t-distribution table can be downloaded from the course website
13
Sample Grades
1 88
2 85
3 90
4 87
5 85
6 80
7 75
8 92
9 95
10 85
75
2
=
=
s
x
Suppose these grades are sampled from a normal distribution with mean μ and standard deviation σ (unknown), find the confidence interval of the mean of the grades with 95% confidence level
t α/2 (10-1) = t α/2 (9) = 2.
The confidence interval for μ is: (86.2-2.262∙5.75/√10, 86.2+2.262∙5.75/√10) = (82.09, 91.31)
α = 0.05, n =
X = grades of student
Exponential Distribution
- Often used to model the time interval between independent
events that happen at a constant average rate
- The probability density function of an exponential function
has the form
14
−
x
e x
f x
x
X
λ λ (Eq. 7)
λ > 0 is a parameter of the distribution, often called the rate parameter
The cumulative distribution is given by
−
x
e x F x
λ x
λ (Eq. 8)
Mean and Variance
E( X ) = 1/λ (Eq. 9) var( X ) = 1/λ^2 (Eq. 10)
16
Suppose the life in hours of a certain type of tube is a random variable that has an
exponential distribution with a mean of 1,000 hours. What is its probability density
function? What is the probability that such a tube will last at least 1,250 hours?
E( X ) = 1000 = 1/λ → λ = 1/
f ( x ) = 1/1000 × e(-1/1000)∙ x
P ( X ≥1250) = 1 - P ( X <1250) = 1- F ( X =1250) = 1- (1- e(-1/1000)∙1250^ ) = e(-1/1000)∙ = 0.
X = life in hours of the tube
