1. Chapter 3-Normal distribution
X : N (, )
Example: N (4,2) , P (X < 6.03)
P (5 < X < 6.03)

2. lognormal distribution
fX(x) x

3. lognormal distribution
If X ~LN (l, z)
lnX ~ N (l, z)

4. Example
1. F-1(0.95) = 1.645
How about the settlement is Log-normal?

5. exponential distribution
fX(x)
x  0

6. Beta distribution x
fX(x)
q = 2.0 ; r = 6.0
probability
a = 2.0
b = 12

7. Standard Beta distribution
fX(x)
(a = 0, b = 1)
q = 1.0 ; r = 4.0
q = r = 3.0
q = 4.0 ; r = 2.0
q = r = 1.0 x
The difference between Beta and other similar distribution

8. Review of Bernoulli sequence model
x success in n trials: binomial
time to first success: geometric
time to kth success: negative binomial

9. Ex 3.54
Statistics show that 20% of freshman in engineering school quit in 1 year. What is the probability that among eight students selected at random, two of them will quit after 1 year?

10. Think: 1. Continuous or discrete?
Students cannot pass or fail “continuously”
Binomial, Geometric or Negative binomial?
Bi: x success in n trials (orderless)
Geo: time to first success (ordered)
Neg: time to kth success (last term ordered)
p = 0.2

11. What is the probability of at least two of them will fail after 1 year?
Use T.O.T:
P (X ≥ 2)
= 1 – P(X = 0) – P(X = 1)

12. what is the probability that among eight students selected at random, two of them will quit within 2 years?
Approach 1: Bayes theorem + TOT
We first consider 1st year scenario:
Why not consider X = 3, 4…...8?

13. For 2nd year: P
= P(0 student in 1st year) P(2 student in 2nd year)
+ P(1 student in 1st year) P(1 student in 2nd year)
+ P(2 student in 1st year) P(0 student in 2nd year)
P = (.167)(.293) + (.335)(.367) + (.293)(.262) = .249

14. Approach 2: Geometric
Recall geometric is “first time to success”, (1-p)t-1p
Students can quit at 1st and 2nd year.
i.e. t=1, t =2
When t = 2, 1st year pass is defined.
P (t = 1) = 0.2 P (t =2) = (0.8) = 0.16
P (a student quit in 1 or 2 year)
= = 0.36