ESS210B
ESS210B
Prof. Jin
Prof. Jin-
-Yi Yu
Yi Yu
Part 1:
Part 1: Probability Distributions
Probability Distributions
Probability Distribution
Normal Distribution
Student-tDistribution
Chi Square Distribution
FDistribution
Significance Tests
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Prof. Jin's ESS210B Lecture Notes: Probability Distributions and Data Analysis, Study notes of Statistics
A portion of Prof. Jin's lecture notes for ESS210B, covering topics such as probability distributions, normal distribution, student-t distribution, chi-square distribution, F distribution, significance tests, organizing data, finding relationships among data, testing significance of results, and using standard normal distribution. The notes also include examples and applications in operational climatology.
What you will learn
- How do you test the significance of the results in data analysis?
- What is the difference between a probability distribution and a normal distribution?
- What is the purpose of organizing data in data analysis?
- What is the role of the standard normal distribution in data analysis?
- How do you find the probability that a value of Z is greater than a certain value?
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2021/2022
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ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Part 1:
Part 1:
Probability Distributions
Probability Distributions
Probability Distribution
Normal Distribution
Student-
t
Distribution
Chi Square Distribution
F
Distribution
Significance Tests
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Purposes of Data Analysis
Purposes of Data Analysis
Organizing Data Organizing Data
Find Relationships among Data Find Relationships among Data
Test Significance of the ResultsTest Significance of the Results
True Distributions or Relationships in the Earths System
Sampling
Weather Forecasts,Weather Forecasts,
Physical Physical
Understanding, Understanding, …
….
.
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Variables and Samples
Variables and Samples
Random Variable Random Variable
: A variable whose values occur at
random, following a probability distribution.
Observation Observation
: When the random variable actually attains
a value, that value is called an observation (of thevariable).
Sample Sample
: A collection of several observations is called
sample. If the observations are generated in a randomfashion with no bias, that sample is known as a randomsample.
By observing the distribution of values in a random sample,we can draw conclusions about the underlying probabilitydistribution.
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Probability Distribution
Probability Distribution
The pattern of probabilities for a set of events is called aprobability distribution. (1) The probability of each event or combinations of events must range from
0 to 1.
(2) The sum of the probability of all possible events must be equal too 1.
continuous probability distribution
discrete probability distribution
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Probability Density Function
Probability Density Function
P
= the probability that a randomly selected value of a variable X falls
between a and b.
f(x)
= the probability density function.
The probability function has to be integrated over distinct limits toobtain a probability.
The probability for X to have a particular value is ZERO.
Two important properties of the probability density function:
(1)
f(x)
0 for all x within the domain of
f.
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Cumulative Distribution Function
Cumulative Distribution Function
The cumulative distribution function
F(x)
is defined as the probability
that a variable assumes a value less than
x
The cumulative distribution function is often used to assist incalculating probability (will show later).
The following relation between F and P is essential for probabilitycalculation:
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Standard Normal Distribution
Standard Normal Distribution
The standard normal distribution has a mean of 0 and a standarddeviation of 1.
This probability distribution is particularly useful as it can representany normal distribution, whatever its mean and standard deviation.
Using the following transformation, a normal distribution of variable Xcan be converted to the standard normal distribution of variable Z:
Z = ( X -
μ
σ
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Transformations
Transformations
It can be shown that any frequency function can be transformed in to afrequency function of given form by a suitable transformation orfunctional relationship.
For example, the original data follows some complicated skeweddistribution, we may want to transform this distribution into a knowndistribution (such as the normal distribution) whose theory andproperty are well known.
Most geoscience variables are distributed normally about their mean orcan be transformed in such a way that they become normallydistributed.
The normal distribution is, therefore, one of the most importantdistribution in geoscience data analysis.
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Another Example
Another Example
Example 2: What is the probability that Z lies between the limits Z
1
and Z
2
Answer:
P(Z
P(Z < 0.60)
Î
P(Z >-0.60)=
P(-0.
Z
0.75) = P(Z
P(Z
negative value
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Example 1
Example 1
Given a normal distribution with
μ
= 50 and
σ
= 10, find the probability
that X assumes a value between 45 and 62. The Z values corresponding to
x
1
= 45 and
x
2
= 62 are
Therefore, P(45 <
X
= P(-0.5 <
Z
= P(Z <1.2) – P(Z<-0.5)= 0.8849 – 0.3085= 0.
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Probability of Normal Distribution
Probability of Normal Distribution
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
Probability of Normal Distribution
Probability of Normal Distribution
There is only a 4.55% probability that a normally distributed variable willfall more than 2 standard deviations away from its mean.
This is the two-tailed probability. The probability that a normal variablewill exceed its mean by more then 2
σ
is only half of that, 2.275%.
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu
How to Estimate
How to Estimate
and
and
In order to use the normal distribution, we need to knowthe mean and standard deviation of the population
Î
But they are impossible to know in most geoscienceapplications, because most geoscience populations areinfinite.
Î
We have to estimate
and
from samples.
and
ESS210BESS210BProf. JinProf. Jin-
-Yi Yu