1. Statistical Distributions

2. Discrete Type
． Bernoulli Distribution
． Binomial Distribution
． Hypergeometric Distribution
． Multinomial Distribution
． Poisson Distribution

3. Continuous Type
． Normal Distribution
． Gamma Distribution
． Exponential Distribution
． Chi-square Distribution
． Student’s T Distribution
． F Distribution

4. ． Bernoulli Distribution
An experiment that can have one(X) of two outcomes:
Success(S, x=1), Failure(F, x=0) Bernoulli experiment
P(S) = P(X=1) = p , P(F) = P(X=0) = 1 - p = q
Probability distribution function

5. ． Binomial Distribution
Repeated n times independent Bernoulli experiment
Binomial experiment
i.e. a Binomial experiment possesses the following properties:
10 the experiment consists of a fixed number n of trials
20 the result of each trial can be classified into one of two categories
30 the probability p of a success remains constant for each trial
40 each trial of the experiment is independent of the other trials

6. Let r.v. X be the number of successes in the n trials of a Binomial
experiment, then X is called the Binomial distribution, .
The probability distribution function is
where
mean
variance

7. Let
be a random sample with Ber(p) and
, then
and

8. ． Hypergeometric Distribution
◆ Sampling with replacement (WR)
v.s.
Sampling without replacement (WTR)
n balls
N balls
sampling
R red balls
Sample
Population

9. Let X be the number of red balls in the sample, then the distribution
of X is the hypergeometric distribution, .
The probability distribution function (PDF) is
mean
variance

10. Theorem: If
, and as
, then for each value
and
with
, a positive constant ,

11. Example:
Ten seeds are selected from a bin that contains 1000 flower seeds,
of which 400 are red flowering seeds, and the rest are of other colors.
10 P(exactly five red flowering seeds)=0.2013
mean = variance = 2.378
20 P(five red flowering seeds by the Binomial approximation)=0.2007
mean = variance = 2.40

12. ． Multinomial Distribution
Extension:
． Multinomial Distribution
If each trial has several different outcomes, label the different possible
types resulting from each trial by i where
the probability of each type at each trial is pi , and the count of each
of the types in a sample of size n as Xi , then the probability of is

13. 寫出 “Extended Hypergeometric Distribution”
H.W.
寫出 “Extended Hypergeometric Distribution”

14. ． Poisson Distribution
the number of cars that are red, out of every 10 cars that pass a
certain spot on a road
Binomial distribution
the number of red cars that pass the spot per hour, without specifying
how many cars in total there are
Poisson distribution

15. r.v.
mean = variance = λ 例:
假設到達某醫院病患人數符合 Poisson 過程,且平均每小時 1
人到達, 則
10 P(1小時內無病患到達)
20 P(1小時內病患到達人數少於4人)

16. Hypergeometric distribution
n/N≦0.05
Binomial distribution
n large, p small(rare event)
Poisson distribution

17. 例:
假設某種疾病治癒率為 2% , 若今有 100 位病患接受治療,
試求最多三人被治癒之機率。
<Sol. Binomial>
令 r.v. X 表治癒人數, 則
<Sol. Poisson>

18. ． Normal Distribution

19. f(x) μ
r.v. X ~ N(μ, σ2)
the pdf for X is

20. X ~ N(μ, σ2) μ
μ-σ
μ+σ
μ-2σ
μ+2σ

21. X ~ N(μ, σ2) a μ b

22. ● X ~ N(μ, σ2) normalized Z ~ N( 0 , 1)
● the pdf for X is z Z ●
(查表)

23. Standard Normal Cumulative Distribution Function Φ(z)
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.5040
0.9463
0.9719
0.9778
0.5080
0.9474
0.9726
0.9783
0.5120
0.9484
0.9732
0.9788
0.5160
0.9495
0.9738
0.9793
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.5239
0.9515
0.9750
0.5279
0.9525
0.9756
0.5319
0.9535
0.9761
0.5359
0.9545
0.9767

24. ● Properties : Z ~ N( 0 , 1) 10
Z ~ N( 0 , 1)
ψ(z) 20 -z z Z
30 X ~ N(μ, σ2)

25. 例:
1. P(– 0.15 ≦ Z ≦1.60) = Φ(1.60) – Φ(– 0.15)
= Φ(1.60) – [1 – Φ(0.15)]
= – [1 – ]
= – =
P(Z ≦ – 1.9 or Z ≧ 2.1)
= P(Z ≦ – 1.9) + P( Z ≧ 2.1)
= Φ(– 1.9) + [1 – Φ(2.1)]
= [1 – Φ(1.9)] + [1 – Φ(2.1)]
= [1 – ] + [1 – ] = =

26. Z ~ N( 0 , 1) α α ● ● 例:

27. 例:
假設某一族群男性體重(X)呈常態分配, 平均體重與標準差分
別是 80 及 5 公斤, 則體重介於 65 和 75 公斤之間的比例有
H.W.
體重超過 85 公斤的比例有多少?

28. ● 10 20
例: Consider sample allele proportions for the ABO blood group
system. For a sample of size 16 alleles from a population in which
allele A has proportion 0.50, the numbers of A are X. 10 20

29. ． Gamma Distribution

30. ． Exponential Distribution
In the Gamma distribution if
then
Theorem: no-memory property

31. ． Chi-square Distribution
In the Gamma distribution if
then

32. 例:

33. ． Student’s T Distribution

34. ． F Distribution

35. 例:

36. Basic Sampling
Distribution Theory

37. ● Statistic is a function of a random sample.
e.q.
● Statistic is a random variable.
Sampling distribution
e.q.

38. Inferential statistics

39. ◆ Estimation of parameter
◆ Testing of statistical hypothesis

40. 估計(Estimation): 由母體抽出樣本,依據樣本統計量的
抽樣分配,推估母體參數真實值。
點估計(point estimation)
區間估計(interval estimation)

41. 點估計(point estimation)
i.e.
Est.
(Statistic)
(Parameter)
e.q.
Est.
Sample mean
Population mean
Est.
Sample variance
Population variance
Est.
Sample proportion
Population proportion

42. 參數的估計是否只存在唯一的估計量(統計量)?
‧ 不偏性(unbiased)
假設 T 為母體未知參數 θ 之一估計量, 若 E(T) = θ,
則 T 為 θ 的不偏估計量(unbiased estimator)
‧ 有效性(efficient)
假設 T1 及 T2 均為母體未知參數 θ 之不偏估計量,
且 Var(T1) < Var(T2), 則 T1 較 T2 具有效性
‧ 一致性(consistant)
假設 T 為母體未知參數 θ 之一估計量,若 ,
則 T 為 θ 的一致估計量(consistant estimator)

43. 例:
r.s.
具期望值 μ , 變異數 σ2 令 則
Unbiased estimator

44. 例:
r.s.
Unbiased estimator
Consistant estimator