1. TOPIC 5
Normal Distributions

2. Start Thinking
As a web designer you face a task, one that involves a continuous measurement of downloading time which could be any value and not just a whole number. How can you answer the following questions:
What proportion of the homepage downloads take more than 10 seconds?
How many seconds elapse before 10% of the downloads are complete? etc.

3. Continuous Probability Distributions
Uniform
Exponential
Normal
Gamma
Weibull
Beta

4. Normal Distribution Also called Gaussian distribution
‘Bell-shaped’ & symmetrical
Mean, median, mode are equal
Random variable has infinite range
Area under the curve is 1 (the probability equals 1) σ
Can be used to approximate discrete probability distributions, for example: Binomial and Poisson
Basis for classical statistical inference

5. Probability Density Function
 = Standard deviation
 = ; e =
x = Value of random variable (– < x < )
 = Mean
The mean and the variance are
The notation that denotes the random variable X has a normal distribution

6. f(x) B A C x Effect of Varying Parameters (μ & σ) 5 10
Normal distributions differ by mean & standard deviation
Each distribution would require its own table.
f(x) B A C x 5 10
That’s an infinite number of table! Then we need a “standardized” normal distribution

7. Standard Normal Distribution
The Standard Normal Distribution
One table!
Normal Distribution
Standard Normal Distribution s
s = 1 m
m = 0 X Z
Negative
Positive

8. Normal Distribution Probability
If X ~ N(μ, σ2) then the transformed random variable is
The random variable Z is known as the “standardized” version of the random variable X.
The probability values of a general normal distribution can be related to the cumulative distribution function of the standard normal distribution, Φ(z)
where

9. Standard Normal Distribution
Example
Normal Distribution
Standard Normal Distribution
s = 1
s = 0.8
m = 3
m = 0 X Z 3
4.2
z = 1.5
Therefore

10. Standard Normal Tables
The values of the cumulative distribution function of the standard normal distribution, Φ(z) or the probability P(Z ≤ z) is already tabulated
m = 0
s = 1 Z
z = ?

11. Normal Distribution Probability
We just have the table of the cumulative distribution function of the standard normal distribution, Φ(z) or P(Z ≤ z) to find P(X ≤ a). By using the same table, we can find the other probabilities m m a a b X X

12. Standard Normal Distribution
Normal Distribution μ = 5, σ = 10 : P(5 < X< 24.6) = ?
Normal Distribution
Standard Normal Distribution
s = 10
s = 1
m = 5
m = 0 5
24.6 X
1.96 Z

13. Standard Normal Probability Table
Normal Distribution μ = 5, σ = 10 : P(5 < X< 24.6) = ?
Standard Normal Probability Table
Look up the table !
0.06 Z
0.04
0.05
1.8
0.9671
0.9678
0.9686
s = 1
1.9
0.9750
0.9738
0.9744
0.4750
2.0
0.9793
0.9798
0.9803
m = 0
2.1
0.9838
0.9842
0.9846
1.96 Z

14. Standard Normal Distribution
Normal Distribution μ = 5, σ = 10 : P(X ≥ 8) = ?
Look up the table ! please
Normal Distribution
Standard Normal Distribution
s = 10
s = 1
m = 5
m = 0 5 8 X
0.3 Z

15. Normal Distribution Example
You work in Quality Control for GE. Light bulb life has a normal distribution with = 2000 hours and = 200 hours. What’s the probability that a bulb will last
between 1800 and 2200 hours?
less than 1470 hours?
more than 2500 hours?
Allow students about minutes to solve this.

16. Standardized Normal Distribution
Example Solution a)
Normal Distribution
Standardized Normal Distribution
s = 200
s = 1
m = 2000
m = 0
1800
2200
-1.0 X
1.0 Z

17. Standardized Normal Distribution
Example Solution b)
Normal Distribution
Standardized Normal Distribution s
= 200 s
= 1
0.0040
m = 2000
m = 0 X
1470
-2.65 Z

18. Standardized Normal Distribution
Example Solution c)
Normal Distribution
Standardized Normal Distribution
s = 200
s = 1
m = 2000
m = 0
2500 X
2.5 Z

19. The Empirical Rule

20. Standard Normal Probability Table
Finding Random Variable X for Known Probabilities
Given that P(X ≤ a) = , what is a?
Standard Normal Probability Table
Firstly, find the value of z ! Z
.00
0.2
0.0
.5000
.5040
.5080
0.1
.5398
.5438
.5478
.5793
.5832
.5871
.6179
.6255
.01
0.3
.6217
.6217 s
= 1 m
= 0
0.31 ? Z
The closest value

21. Standard Normal Distribution
Finding Random Variable X for Known Probabilities
Secondly, find the value of X = a !
Normal Distribution
Standard Normal Distribution s
= 10 s
= 1
.6217
.6217 m
= 5
8.1 ? X m
= 0
0.31 Z

22. Exercise
The thicknesses of metal plates made by a particular machine are normally distributed with a mean 0f 4.3 mm and a standard deviation of 0.12 mm
What is the proportion of the metal plates that have thickness outside the range of 4.1 to 4.5 mm
What are the upper and lower quartiles of the metal plate thickness?
What is the value of c for which there is 80% probability that a metal plate has a thickness within the interval [4.3 – c, c]?

23. Answer to the Exercise μ = 4.3 mm and σ = 0.12 mm a)
b) Lower quartile: P(X ≤ a) = and upper quartile: P(X ≤ a) = 0.75

24. Answer to the Exercise
c) It means P (a ≤ X ≤ b) = 80%, where P (X ≤ a) = 10% = 0.1 or P (X ≤ b) = 90% = 0.9. Pick either one.

25. Linear Combination of Normal Random Variables
Linear Functions of a Normal Random Variable
If X ~ N(μ, σ2) and a and b are constants then
The Sum of Two Independent Normal Random Variables
If X1 ~ N(μ1, σ12) and X2 ~ N(μ2, σ22) are independent random variables then
Averaging Independent Normal Random Variables
If Xi ~ N(μ, σ2), 1≤ i ≤ n, are independent random variables then their average is distributed

26. Example
The annual return of the stock of company A, XA say (in percent), is distributed
In addition, suppose that the annual return from the stock of company B, XB say, is distributed
independent of the stock of company A.
What is the probability that company B’s stock performs better than company A’s stock?
What is the probability that company B’s stock performs at least 2% points better than company A’s stock?
Allow students about minutes to solve this.

27. Example Solution Let Y = XB – XA , then Performs better means Y ≥ 0.
Allow students about minutes to solve this.

28. Example Solution b) It means Y ≥ 2.0.
Allow students about minutes to solve this.

29. Normal Approximations to the Binomial Distribution
Not all binomial tables exist
Requires large sample size
Gives approximate probability only
Need correction for continuity
n = 10 p = 0.50 .0 .1 .2 .3 2 4 6 8 10 x
f(x)
The distribution B(n, p) can be approximated by a normal distribution with the mean and variance

30. x a Normal Approximations to the Binomial Distribution f(x) .3 .2 .1
Probability Added by Normal Curve .3 .2
As the number of vertical bars (n) increases, the errors due to approximating with the normal decrease. .1 .0 x a
Binomial Distribution: the area of all the ‘orange’ bars
Normal Approximation: the area starting from the ‘blue’ vertical line to the left. So it needs correction of a ‘half’ in order to have the same area as the Binomial

31. a Correction for Continuity -0.5 +0.5
A 1/2 unit adjustment to discrete variable
Improves accuracy
Correction for each of four cases:
For P(X ≥ a), use the area above â = (a – 0.5).
For P(X > a), use the area above â = (a + 0.5).
For P(X ≤ a), use the area below â = (a + 0.5).
For P(X < a), use the area below â = (a – 0.5). a
-0.5
+0.5

32. Normal Approximation Procedure
Normal approximations to the binomial distribution work well as long as
For each of four cases above, use
where

33. Example
Suppose that a fair coin is tossed n times. The distribution of the number of heads obtained, X, is B(n, 0.5). If n = 100, what is the probability of obtaining between 45 and 55 heads? are satisfied since

34. Example Solution
Using a statistical software or Excel, the exact solution of the binomial probability is The difference is just about

35. Central Limit Theorem
If X1, …, Xn is a sequence of independent identically distributed random variables with a mean μ and a variance σ2 (not necessarily normal distributed), then the distribution of their average X can be approximated by a
distribution. Similarly, the distribution of the sum X1 + … + Xn can be approximated by a
distribution. The general rule is that the approximation is adequate as long as n ≥ 30

36. Any Questions ?