Biostatistics (생물통계학)

Lecture 8

Probability distributions

Doo-Hyung Lee, Department of Life Sciences

http://www.vias.org/science_cartoons/thumbnails.htmlhttp://www.vias.org/science_cartoons/thumbnails.html

Normal Distribution

Normal Distribution

우리의 관심은???

Normal Distribution

우리의 관심은???

주사위를 던졌을 때와는 틀림.

Normal Distribution

우리의 관심은???

주사위를 던졌을 때와는 틀림.

관심대상 변수가 어떠한 구간에 존재할 확률?

Normal Distribution

Standard Normal Distribution

0 -1 -2 -3 3 2 1

Summary for Mid-term Exam

Populations vs. Samples

Population (parameter)

Sample (statistic)

𝜇𝜇 = ∑ 𝑋𝑋𝑖𝑖𝑁𝑁𝑖𝑖=1𝑁𝑁

𝑋𝑋 = ∑ 𝑋𝑋𝑖𝑖𝑛𝑛𝑖𝑖=1𝑛𝑛

𝜎𝜎2 = ∑ (𝑋𝑋𝑖𝑖−𝜇𝜇)2𝑁𝑁𝑖𝑖=1

𝑁𝑁 𝑠𝑠

2 = ∑ (𝑋𝑋𝑖𝑖−𝑋𝑋)2𝑛𝑛𝑖𝑖=1

𝑛𝑛 − 1

Mean:

Variance:

I’ve summarized my data using the frequency table.

Plants / Quadrat ( Xi )

Frequency ( fi )

0 268

1 316

2 135

3 61

4 15

5 3

6 1

7 1

050

100150200250300350

0 1 2 3 4 5 6 7

Plants / Quadrat ( Xi )

Freq

uenc

y ( f i

)

http://companionplants.com/catalog/index.php

Frequency table vs. Box plot

Plants / Quadrat ( Xi )

Frequency ( fi )

0 268

1 316

2 135

3 61

4 15

5 3

6 1

7 1

Probability Density Function

Probability Distributions

Probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference

Random variable is a variable whose actual value is determined by chance operations. A discrete random variable can assume only certain values, either finite or countably infinite. A continuous random variable can assume any value within an interval or intervals of real numbers.

Famous Probability Distributions

• Binomial Distribution

• Poisson Distribution

• Normal Distribution

• Standard Normal Distribution

𝝁𝝁 = 𝑬𝑬 𝑿𝑿 = �𝒙𝒙𝒙𝒙(𝒙𝒙)

𝝈𝝈𝟐𝟐 = 𝑬𝑬 𝑿𝑿𝟐𝟐 − 𝑬𝑬 𝑿𝑿 𝟐𝟐

Situation

f(x)

Binomial Distribution

우리의 관심은???

주사위를 던졌을 때와는 틀림.

성공이 몇번이나 나오는지가 관심! 그때의 확률이 궁금!

Binomial Distribution

𝜇𝜇 = 𝑛𝑛𝑛𝑛 𝜎𝜎2 = 𝑛𝑛𝑛𝑛(1 − 𝑛𝑛)

Famous Probability Distributions

• Binomial Distribution

• Poisson Distribution

• Normal Distribution

• Standard Normal Distribution

𝝁𝝁 = 𝑬𝑬 𝑿𝑿 = �𝒙𝒙𝒙𝒙(𝒙𝒙)

𝝈𝝈𝟐𝟐 = 𝑬𝑬 𝑿𝑿𝟐𝟐 − 𝑬𝑬 𝑿𝑿 𝟐𝟐

Situation

f(x)

Poisson Distribution

우리의 관심은???

주사위를 던졌을 때와는 틀림.

관심 사건이 주어진 시/공간 에서 몇번이나 발생하는지

그때의 확률이 궁금!

Poisson Distribution

• Suppose you typically get 4 pieces of mail per day.

• Sometimes a little more, sometime a little less.

• The Poisson Distribution will tell you how likely it is that you will get 3, or 5, or 11 or any other number.

Poisson Distribution

𝒙𝒙 𝒙𝒙 = 𝒆𝒆−𝝁𝝁(𝝁𝝁)𝒙𝒙

𝒙𝒙!

𝐸𝐸 𝑋𝑋 = 𝜇𝜇 𝜎𝜎2 = 𝜇𝜇

http://www.google.co.kr/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&docid=EYOb2nGU9ZAPXM&tbnid=XLkAG36MXaB-jM:&ved=0CAUQjRw&url=http://www.wildco.com/Hand-Held-Tally-Counter.html&ei=7WIzU9XoN4yClQXp1IC4DA&bvm=bv.63738703,d.dGI&psig=AFQjCNH4oGeDCqEuBRDMdx7UZrbx8l-yvA&ust=1395962989099369

Probability Distributions

Probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference

Random variable is a variable whose actual value is determined by chance operations. A discrete random variable can assume only certain values, either finite or countably infinite. A continuous random variable can assume any value within an interval or intervals of real numbers.

Normal Distribution

우리의 관심은???

주사위를 던졌을 때와는 틀림.

관심대상 변수가 어떠한 구간에 존재할 확률?

Normal Distribution

Standard Normal Distribution

0 -1 -2 -3 3 2 1

Biostatistics �(생물통계학)Normal DistributionNormal DistributionNormal DistributionNormal DistributionNormal DistributionStandard Normal DistributionSummary for Mid-term ExamPopulations vs. SamplesI’ve summarized my data using the frequency table.Frequency table vs. Box plotProbability Density FunctionProbability Distributions슬라이드 번호 14Famous Probability DistributionsBinomial DistributionBinomial DistributionFamous Probability DistributionsPoisson DistributionPoisson DistributionPoisson DistributionProbability DistributionsNormal DistributionNormal DistributionStandard Normal Distribution