ฝึกพูดภาษาอังกฤษด้วยเทคนิค Shadowing จากวิดีโอ: 1. Introduction to statistics - Lecture 1

Dear students, welcome.

My name is Ludovico Biagi, I am professor in Politecnico di Milano.

And with this slot of lectures I would provide you a general introduction to statistics

and probabilities and what it is needed to better understand more complicated ideas relevant to geospatial intelligence.

The slot of lectures on statistics will last about more or less five hours.

It is split in several slots and each one's slot concerns a specific topic

and each one's slot is accompanied by exercises that you can do by yourself.

and questions to let you understand if you have completely understood what is contained in the lectures.

First of all, I would start with a general outline about the following slides that we will see in the next slots.

We will start with the general definition of probability, the general properties of probabilities,

then we will introduce the idea of conditional probability

and this leads to the definition of the theorem of bias

and the idea of total probability and then we will move to the concept of independent events.

After the discussion of probability we will move toward the idea of random variables.

We will start in one dimension, so we will define what is a random variable,

we will define what is the distribution and the density functions of a random variable, we will see some specific examples of random variables,

because these examples are very important for the following.

We will define what is the function of one random variable, and then we will see what is the mean and the variance of a random variable.

Then we will move to random variables in two dimensions, I mean x and y.

We will define them, we will see what is the joint density function,

what are the marginal distributions of the two components of a random variable in two dimensions.

again what is the mean the covariance, the propagation laws for the mean and the covariance,

the conditional distributions and the concept of independence between the two components of a random variable in two dimensions.

The next slot will be about the generalization to Rn dimensions.

so again we will see how the properties of random variables in two dimensions can be extended to a general n-dimensional space.

We will see also the differences in Rn with respect to R2.

And finally, a slot will be devoted to the so-called asymptotic behavior of random variables

that puts a very important link between the theory of probability and the practice of observations.

I mean why and how the general theory of probabilities

and random variables can be applied in the management and analysis of observations.

Finally, we will see a particular example of estimator for random variables, that is least squares.

So we will put the general position of the problem, we will see what are for least squares the estimation principles,

what are the estimates provided by least squares, both of the unknowns and of the covariances of the estimated parameters.

Then in the last, very last part, we will see what does it mean to make an hypothesis testing

and in particular we will see how to apply this general principle to the global model

and to the individual observations in least squares.

So let's start with the probability and in the order we will see the axiomatic definition of probability,

the properties of a probability, what is the conditional probability,

what is the total probability of one event and what is the bias theorem.

We are not going to prove theorems just to enunciate and discuss them and finally we will see what are independent events.

So in general what are stochastic experiments and what is their modeling?

Let's start with the definition of a probability model, we can call it capital omega.

The elements of a probability model are called also experimental outcomes and we will denote them with the lower case omega.

The subset of capital omega are named events.

In this picture you see that in orange it is the complete set,

so capital omega, and the white area is a specific event,

capital A, named capital A.

Just to complete the model we need clearly to assign probabilities to all the events capital A

that compose the capital omega set.

So we can start with the axiomatic definition of probability.

Let's take an event capital A.

The probability of capital A is a certain number

that we can symbolize with the p of capital A that is related to capital A.

This number must satisfy the following properties axioms.

The probability of an event is not negative, so it is bigger or equal to zero.

The probability of the whole set of events, so capital omega, is equal to one.

Then let's take two exclusive events, capital A and capital B,

then the probability of their union, the probability that they happen together,

is given by the probability of A plus the probability of B.

Let's generalize the idea of the tiered axiom.

By looking to the picture we have again the capital omega set of events.

Let's suppose you have three events in this case A, B and C that are depicted with the white, blue

and red areas, the probability of A union B union C is equal to

the probability of A plus the probability of B plus the probability of C.

This can be extended to any finite number of terms

and this could be extended also for an infinite but countably many terms.

This is the axiom of infinite additivity that states that the union,

the probability of the union of an infinite number of events a1, a2, etc, etc, etc,

is simply given by the sum of probability of a1 plus

probability of a2 plus up to an infinite sum on all these set of events.

More and more about properties.

The probability of the impossible event, or if you prefer the zero event, is equal to zero.

Moreover, let's look to the picture.

You have capital omega, that is the whole set of possible events.

You have the event A, again the white area.

The complementary of A is defined as omega minus A, so all the events that are not A,

and the probability of the complementary of A is 1 minus the probability of A.

Now the probability of an event A is always between 0 and 1,

0 for the impossible event and 1 for capital omega taken as a wall.

Let's consider two events, B and A, and let's suppose that B is a subset of A.

In this case, the probability of B is smaller or equal to the probability of A.

Now let's suppose A and B, on the contrary, are two events with an intersection exactly in the picture.

You see there is an orange omega, in white A, in blue B, and you see that they have an intersection.

In this case the probability of A union B is exactly

equal to the probability of A plus the probability of B minus the probability of their intersection.

An experiment, capital omega, is specified in terms of the probabilities of all the events that compose the experiment.

However, considering the previous axioms, we don't need to assign probabilities to every event.

In particular, if capital omega consists of n outcomes,

the experiment is specified in terms of the probabilities of the elementary events lowercase omega.

The foregoing can also also, if capital omega consists of a countable number of outcomes.

On the contrary, it doesn't hold when the elements of capital omega are not countable,

for example the real valued points of an interval that is not countable.

Let's consider an experiment that can output a certain number capital N of outcomes.

These outcomes are said to be equally likely

when they have exactly the same probability and each one probability is exactly equal to 1 over capital N.

Now let's suppose that an event consists of NA outcomes.

Then the probability of A can be simply written as Na over N.

As an example let's consider the classical coin or the classical die.

We can say that the coin is fair when its outcomes are equally likely,

100

that is, the probability of heads is equal to the probability of tails and they are both equal to 1 over 2.

101

and let's look to a die, a die has six possible outcomes from one to six,

102

they are equally likely when probability of one is equal to probability of two up to probability of six,

103

and this clearly means that each one probability is one over six now let's suppose you toss a coin twice,

104

so you can have four different possible outcomes, events.

105

Indeed, the capital omega is composed by a head head,

106

head tail, tail head and tail tail.

107

If these four outcomes are equally likely, then each one of them has probability equal to 1 over 4.

108

Now let's suppose that you want to check the probability of the event A that both the second toss,

109

pardon, has outcome as tail.

110

This event A is generated by age T or TT.

111

So two possible outcomes generate the event and so the probability of the event is 1 over 2.

112

Now let's suppose that our set capital omega consists not of a number countable of outcomes

113

but of a non-countable number of outcomes.

114

For example the possible points in the real line, in this case capital omega is defined from minus infinite to plus infinite.

115

In this case the experiment can be specified not in term of a probability but in term of a density function.

116

Density function alpha is named in this slide is always bigger

117

or equal to zero and integral between minus infinite and plus infinite of alpha is equal to 1,

118

and in particular alpha is such that the probability

119

that an outcome is comprised between a minimum value a

120

and the maximum value b is given by the integral from a to b of alpha in dt.

121

Now we can move to the idea and the properties of the conditional probability.

122

Let's take any event M such that the probability of M is bigger than zero,

123

strictly bigger than zero, and let's give another event A.

124

We can compute the ratio probability of A intersection M divided by probability of M.

125

You see in the picture in the bottom of the slide the example you have capital you have m in red,

126

you have a in gray and they have an intersection.

127

The ratio we have just defined is called a conditional probability of a assuming

128

that m happened and it is denoted by the symbol probability of a given m

129

and okay and it is the ratio of p intersection m divided by m.

130

The meaning of conditional probability can be simplified as follows.

131

You know that an event m happened,

132

you just ask what is the probability that given m, a happens.

133

The properties of the conditional probability are the following.

134

Let's take again an event M

135

and as a range as a ranges over all the events of capital omega we have

136

that the probability of A given M is always bigger

137

or equal to zero the probability of omega given M is equal to 1,

138

and in particular, they take two different events, A and B.

139

In case the intersection between A and B is void, we can write

140

that the probability of A union B given M is the

141

probability of A given m plus the probability of b given m.

142

For example, let's take again the example of a die.

143

We remember that the possible outcomes of a die are six, exactly from one to six.

144

we want to determine the probability to obtain the outcome 2 assuming that an even outcome is given.

145

So the probability of M, probability of even, is 1 over 2.

146

The probability of 2 given even is the probability of A intersection M divided by the probability of M.

147

The probability of A intersection M is 1 over 6, the probability of M is 3 over 6,

148

so the probability of A given M is 1 over 3.

149

In the picture in the bottom of the slide you see exactly the example, capital omega,

150

m that is the set of even outcomes 2 4

151

and 6 and a that is the only one outcome equal to 2.

152

Now let's move to the concept of total probability

153

and the theorem of bias that is very important for many many many applications.

154

Now let's suppose to have a partition of capital omega.

155

I mean that we have a set of events a1, a2, up to a m,

156

such that the intersection between ae and aj is zero for each couple of ej,

157

and the union of all the events a is the whole capital omega.

158

Now let's suppose that you have an event B.

159

The probability of B can be written as the probability of B given A1

160

that multiplies the probability of A1 plus the probability of B

161

given A2 multiplied by the probability of A2 summed up to the whole set of events A,

162

so summed up to AM.

163

So let's see the idea with a simple example.

164

We have a political poll and we have recorded the following results.

165

Among all the voters, 70% are male and 30% are female.

166

Among males, 40% are Republican and 60% are Democrats.

167

Among families 45% are Republican and 55% are Democrats.

168

You need to find the probability that a voter selected at random is Republican.

169

So from a graphical point of view, you see in the picture the set capital omega all the

170

votes the orange set is split by a line into male

171

and female voters and the outcomes are split in a set

172

that is again orange of republicans and a red set of Democrats that are clearly shared between families and males.

173

So the probability of a male voter is 0.7,

174

the probability of a family voter is 0.3, the probability of a Republican given

175

m is 0.4 the probability of R given F is 0.45

176

so the probability of R is the probability of R given

177

M multiplied by the probability of M plus the probability of R given F multiplied by the probability of F.

178

If you make the computation, you find that it is equal to 0.415, that is 41.5%.

179

Now a very important theorem that we are not going to prove

180

but just to discuss let's suppose that you have a partition of capital omega

181

and we call this partition again a1 a2 up to a m it means

182

that the their intersection is void for each couple

183

and their union is the wall omega now let's suppose

184

that you have another event b

185

that in the picture is depicted as the red area

186

and the probability of a given a given b so probability of ae given b

187

is equal to the probability of b given ae

188

that multiplies the probability of ae divided by the sum of probability of b given ae

189

that multiplies probability of ae summed on all the a events,

190

so summed on all the m a events.

191

And with the help of two coins we can depict an example.

192

Let's suppose we have two coins a and b.

193

Coin a is fair, so the probability of head is equal to the probability of tail

194

while coin of b is loaded with a probability for s

195

that it is 2 over 3

196

so probability of tail for b is 1 over 3 let's

197

suppose you pick up one of the coin at random without knowing

198

if it is a or b let's suppose you toss it

199

and the outcome is s what is the probability that we picked the fair coin.

200

So the probability can be computed as probability of F given add,

201

so probability of add given A

202

that multiplies probability of A over the probability of edge given A

203

that multiplies probability of A plus probability of edge given B multiplied by probability of b.

204

You make all the computations and you find that the probability is 3 over 7.

205

Now let's move to the idea of independent events.

206

We have two events, a and b.

207

They are called statistically independent when the probability of their intersection is exactly equal to the product of their individual probabilities.

208

So from a mathematical point of view we can write

209

that probability of A intersection B is equal to probability of A that multiplies probability of B.

210

What happens when two events are statistically independent for conditional probabilities?

211

okay the probability of A given B

212

that is defined as the probability of A intersection B divided

213

by probability of B is equal to probability of A

214

that multiplies probability of B divided by probability of B

215

that is clearly equal to probability of A in the same way we see

216

that probability of B given A is simply given by probability of B.

217

Again let's help with a simple example.

218

Let's suppose we have a fair coin and a fair die and let's suppose we toss both of them.

219

Clearly the two experiments are independent.

220

So let's suppose we want to compute the probability

221

that 5 shows on the die and the output of coin is 8

222

so the probability of the

223

that the two outcomes happen together is the probability of the intersection

224

that in this case because of the independence is the product of the individual probabilities

225

and so we have 1 over 12 so this ends

226

this first set of slides about statistics and at the end we find questions

227

that allow you to self-evaluate yourselves i mean

228

if you are able to answer to the questions you have

229

understood the ideas on the contrary case it is better you try to look again to the lecture

230

so the first question is what is a stochastic experiment outcome

231

and what is an event the second question is what is

232

the set capital omega of a probability space the third question

233

is what is the set how it is defined the set of all the possible events

234

and what are the properties of this set now you should be able to give the axiomatic definition of probability

235

and now let assume

236

that the set omega is discrete is it sufficient to assign

237

the probability to each outcome to define it why it is the answer give an example of uniform distribution of probability

238

possibly not the coin or the die, if the set capital omega is continuous,

239

how we can specify the probability of an event,

240

give the definition of conditional probability and the definition of of stochastically independent events,

241

what is a partition of capital omega, and discuss the bias theorem.

242

Moreover, in the exercises folder of this set of lectures,

243

you will find a few very simple exercises that you should try to solve by yourself.

244

Thank you.

✨ วิดีโอแนะนำ

36 ประโยค

สิ่งที่คุณจะได้เรียนรู้

จากวิดีโอนี้ นักเรียนจะได้เรียนรู้เกี่ยวกับพื้นฐานของสถิติและความน่าจะเป็น ซึ่งจะช่วยให้เข้าใจแนวคิดที่ซับซ้อนมากขึ้นที่เกี่ยวข้องกับข้อมูลทางภูมิศาสตร์ ในการเรียนรู้ครั้งนี้ คุณจะได้ฝึกทักษะการพูดที่สำคัญสองสามอย่าง เช่น การใช้คำศัพท์เฉพาะที่เกี่ยวข้องกับความน่าจะเป็น และการตีความข้อมูลที่เกี่ยวข้องกับเหตุการณ์สุ่มที่จะช่วยสร้างความมั่นใจในการสื่อสารของคุณในบทสนทนาต่าง ๆ

ฟังเสียงที่ต้องใส่ใจ

ในวิดีโอนี้ มีการใช้ลิงกิ้งและการลดเสียงที่ทำให้เสียงธรรมชาติของการพูดมีความไหลลื่นมากขึ้น ตัวอย่างเช่น การพูดคำว่า "probability model" จะมีการเชื่อมเสียงระหว่างคำทำให้เสียงฟังดูนุ่มนวลมากขึ้น การฟังและสังเกตเสียงเหล่านี้จะช่วยให้คุณเข้าใจวิธีการพูดที่เป็นธรรมชาติของเจ้าของภาษาได้ดียิ่งขึ้น ซึ่งเป็นวิธีหนึ่งในการพัฒนา "shadowspeaks" ของคุณ เมื่อคุณเรียนรู้ที่จะฟังเชื่อมระหว่างคำ คุณจะแสดงออกในภาษาอังกฤษได้อย่างมีประสิทธิภาพมากขึ้น

พูดอย่างไรให้เหมือนเจ้าของภาษา

การทำซ้ำเสียงในวิดีโออย่างมีจังหวะจะช่วยให้คุณสามารถจับจังหวะและน้ำเสียงของผู้พูดได้ การวางน้ำหนักเสียงในจุดที่สำคัญ เช่น การเน้นที่คำว่า "probability" หรือ "random variable" จะทำให้การพูดของคุณมีพลังยิ่งขึ้น การฝึกพูดตามจังหวะนี้คล้ายกับการทำ "shadow speech" ช่วยให้คุณเข้าใจการใช้เสียงสูงต่ำในภาษาอังกฤษได้ดียิ่งขึ้น นี่จะเป็นประโยชน์ต่อการพัฒนาทักษะการพูดในภาษาอังกฤษของคุณ และทำให้การเรียนรู้ผ่านวิดีโอเป็นเรื่องสนุกและมีประสิทธิภาพ

ในการพัฒนาทักษะการพูด คุณสามารถใช้เทคนิคนี้ใน "shadowing site" ที่เสนอเนื้อหาบทสนทนาและการสนทนาในหัวข้อต่าง ๆ เพื่อให้การเรียนรู้ของคุณเป็นไปอย่างต่อเนื่องและมีประสิทธิภาพมากขึ้น

ไวยากรณ์ในวิดีโอนี้

โครงสร้างที่ผู้พูดใช้บ่อยที่สุด พร้อมคำพูดจริงจากวิดีโอ:

โครงสร้าง	ในวิดีโอ
Passive voice be + กริยาช่อง 3 — เน้นสิ่งที่เกิดขึ้น ไม่ใช่ผู้กระทำ	is needed · is split · is accompanied
Present perfect have/has + กริยาช่อง 3 — เหตุการณ์ในอดีตที่ยังเกี่ยวข้องกับปัจจุบัน	have just defined · have recorded

เทคนิค Shadowing คืออะไร?

Shadowing เป็นเทคนิคการเรียนรู้ภาษาที่ได้รับการรับรองทางวิทยาศาสตร์ พัฒนาขึ้นสำหรับการฝึกนักแปลมืออาชีพ วิธีการนี้เรียบง่ายแต่ทรงพลัง: คุณฟังเสียงภาษาอังกฤษจากเจ้าของภาษาและพูดตามทันที — เหมือนเงาที่ตามผู้พูดด้วยช่วงเวลาห่าง 1-2 วินาที การวิจัยแสดงว่าเทคนิคนี้ปรับปรุงความแม่นยำในการออกเสียง ทำนองเสียง จังหวะ การเชื่อมเสียง การฟังเข้าใจ และความคล่องแคล่วในการพูดได้อย่างมีนัยสำคัญ

เทคนิค shadowing: อ่านคู่มือฉบับเต็มทีละขั้นตอน →

ฝึกพูดภาษาอังกฤษด้วยเทคนิค Shadowing จากวิดีโอ: 1. Introduction to statistics - Lecture 1

An introduction to a lecture

Lec 1 | MIT 18.06 Linear Algebra, Spring 2005

What is NOT Random?

What is Statistics? A Beginner's Guide to Statistics (Data Analytics)!

Introduction to Hematology

Contemporary Topics 2 Unit 1 Lecture with Powerpoint

How AI works: Machine learning

The Map of Physics

1 8 What is Causation

Eco 155: Principles of Macroeconomics Class 1

Introduction to Cognitive Bias: Crash Course Scientific Thinking #1

RE Unit 4B

สิ่งที่คุณจะได้เรียนรู้

ฟังเสียงที่ต้องใส่ใจ

พูดอย่างไรให้เหมือนเจ้าของภาษา

ไวยากรณ์ในวิดีโอนี้

เทคนิค Shadowing คืออะไร?