Link to my example of when i used to be a child and visual intuition vs the other way round.

Statistical Independence: Homework and Teacher Collection

Understanding Independence Through Examples

Three Scenarios of Dependence/Independence

Scenario 1: Perfect Positive Dependence

• Each time you don’t do homework, the teacher forgets to collect it
• The events guarantee each other - they are perfectly dependent
• If you know one event, you can predict the other with 100% certainty

Scenario 2: Perfect Negative Dependence

In mathematics one start from definitions and axioms and derive useful truth about those mathematical object. The definition of distance in mathematics serves as the perfect oppurtunity to concretize this system:

Pick any two numbers, x y their distance d(x,y) equals absolute x - y

ABsolute value on the other hand is defined as give definition piecwise of absolute value

If you have 40 on the first midterm 60 on the second midterm exam. You improved by +10 mark. The absolute value of your increase is 10. If you get 50 on the final your grade changed by -10 marks but the absolute value of your change in grade is absolute -10 = 10. Absolute strips of

Result number 1 : absolute value is non negative. As we can see from the definition. It will absolute value takes a negative number it does a procedure that always change it to positive

If x = 0 absolute value must be zero. and vice versa

By how we defined it if x = 0 aboslute value turns into 0. Think of it like zero is neutral. BUt absolute value only cares about those negative signs. it elimanates it.

on the other hand if we see absolute x = 0 what can we say about the original x?

There are three types of number positive, negative and zero? ANd each is exactly of that categeroy. So If you think of the absolute value as a machine, if it spits a zero, could the number he received be a positive? The absolute value machine does noting to the positive number it takes, So if you enter positive you can come out negative. If you enter negative you come out positive. The only way to come out zero is to enter zero. If anything is positive consider. If a zero comes out of that aboslute value machine you know what entered.

Abslute value of x smaller or equal to y gives a new interval with a lower bound of -y and upper of y.

You’re stuck in a desert. At point x Draw the line. Given the heat you can only survives absolute value kilometer without dying. If you go straight ahead we consider that you walked . If you go back to where you came from you walked - backward. If I give you . Then can you reach the any of the oasis?

A picture showing the situation

Link it to the stateement

Triangle inequality

From adrenaline you can muster the strength to walk 2 more steps after you walk your 2 meters. If you walk your 3 steps forward then 2 steps forward In the postive direction you are guaranteed to reach the oasis. If someone on the other end walk 3 steps forward but used their 2 steps to go back again can they reach? the oasis? That’s what is encapuslated in the triangel property. Walking positively will always lend you the farthest right? on the number line. Compared to adding number

Now how to draw a triangle from 2 sides lenght 2 and 3 remove them. If you place them one after the other can you create a triangle?

Show a picture that show them that you can draw a third sides to form a triangle?

What about if create an angle from the line now you can create tirangel ?

Show a lot o f bent angle creating different triangle.

All those bents are shorter compared to our third sides showing a picture of that. SO 2 sides of a triangle must always be greater or equal to the third sides in lenght. Notice that we use distance and lenght interchangeably that’s wny this theorem is called triangle inequality

If you gain 100000 you end up with positive 100000. But if you spend that 100000 three times you balance end up at -100000 . But the absolute value in both of those multiplication is 300000. Justifying the stateement Show that statement about mutiplying absolute value

Absolute value of distance

What’s distance as a computation that gives you the lenght between any point on the number line

TIfe

My salary is twice yours. If your salary is x. Where is a number. Mine is 2 times x. Let’s called mine y. y = two times x.

x and y are called variable. They allow us to captures relationships without needing exact numbers. We can graph relationship using the cartesian plane:

Plot of that relationship in hugo.

What if I introduced a new variable called u. u = x . so I can write the function y = x as y = u. Here is the new graph.

Understanding Averages: From Student Grades to Speed Calculations

Quiz Results Analysis

In a class of five students, here are the results of a quiz on 10 marks:

• $s_1 = 8$
• $s_2 = 6$
• $s_3 = 9$
• $s_4 = 4$
• $s_5 = 7$

What’s the average performance in this class?

You do the total number of marks (calculation) divided by the total number of students:

$$\text{Average} = \frac{8 + 6 + 9 + 4 + 7}{5} = \frac{34}{5} = 6.8$$

Explain a bernoulli Event

Explain how to calculate a bernoulli and what it gives

Explain that x_n is increasing the number of time you consider that event

Explaining that increasing the x_n makes the data approaches a normal distribution