revision: complex number

Revision: Binomial, Poisson and Normal distribution

revision: integration

Revision: Differentiation

Lecture 4: Normal distribution

In today's lecture, we learnt two approximations.

1. Binomial distribution to normal distribution

If X~B(n, p)

• Recall that if when n is large, and np is smaller than 5, we can approximate X to Poisson distribution with mean np, i.e. X~Po(np)
• If n is large, np is larger than 5 and n(1-p) is also larger than 5, we can approximate X to Normal distribution with mean np and variance np(1-p), i.e. X~N(np, np(1-p)

2. Poisson distribution to normal distribution

If \[X \sim Po(\lambda)\], and \[\lambda>5\],

we can approximate X to normal distribution, \[X \sim N(\lambda,\lambda)\]

Remark: Remember to do the continuity correction when you approximate either Binomial or Poisson distribution to Normal distribution.

You can attempt tutorial 15B now.

Lecture 3: Normal distribution

In today's lecture, we learnt three important properties of Normal distribution.

1. Sum of two independent normal random variables

2. Sum of n independent observations from the same Normal distribution

3. Multiple of Normal variables.

Remember the following rules:

• All the random variables must be independent.
• When two random variables are added or subtracted, their variance are always added.
• When a random variable is multiplied by a constant, say, a, the variance is multiplied by a^2.
• You can only add or multiply the variance, NOT the standard deviation.

Lecture 2: Normal distribution

In this lecture, you have learnt the following points:

1. the use of " invNorm"

Example: X~ N(17,11), P(X is smaller than a ) =0. 38, use invNorm (0.38, 17, sqrt(11)) to find the value of a.

Note: If the given information is P(X is bigger than a)=0.38, you have change it to P(X is smaller than a)=1-0.38=0.62 before you use the "invNorm" function.

2. Properties of Normal distribution

If \[X\sim N(\mu ,\sigma ^{2}) \], \[aX+b \sim N(a\mu+b, a^{2}\sigma^{2})\]

3. Standard Normal distribution

• Standard Normal distribution is the normal distribution with mean 0 and variance 1.
• We can change any normal distribution to standard normal distribution. \[X\sim N(\mu ,\sigma ^{2}) \], \[\frac{X-\mu}{\sigma}\sim N(0,1)\]

You can make use the properties of Normal distribution to prove it.

Binomial and Poisson distribution Lecture 5+ Normal distribution Lecture 1

1. Binomial and Poisson distribution

For binomial and poisson distribution, we learnt about the approximation binomial distribution to poisson distribution today.

If X~ B(n, p), with n is larger than 50, p is smaller than 0.1 with np smaller than 5, X~ Po(np) approximately.

2. Normal distribution

In today's lecture, we learnt a few things on Normal distribution.

• Continuous random variable

Normal distribution is for continuous random variables. ( Unlike Binomial and Poisson distributions, which are for discrete random variables. )

For a continuous random variable, there is a probability density function (pdf) f(x) associated with it.

However, f(a) does not denote the probability at a. Instead, for continuous random variable, the probability at any specific value is 0. We can only compute the probability of a certain interval, e.g. (a, b). The area under the curve of f(x) from a to b will give the probability P(x is between a and b )

• Important result for Expectation and variance

Expectation

E(a)=a

E(aX+b) = aE(X) + b

E(aX+bY)=aE(X)+bE(Y)

Variance

Var(a) = 0

Var(aX+b)=a^2 Var (X)

Var(aX+bY)= a^2 Var(X)+b^2 Var(Y) if X and Y are independent

• Normal distribution \[X\sim N(\mu, \sigma ^{2})\]

Normal distribution is symmetrical about the mean. The mean, median and mode are all equal.

When you use G.C. to evaluate normal distribution, you should key in:

normalcdf ( lower bound, upper bound, mean, standard deviation)

Lecture 4:Binomial and Poisson distribution

In this lecture we learnt two important ideas.

1. Find the mode for Poisson distribution

• Similar to Binomial distribution, the Poisson distribution has probabilities that increase to a certain level and decrease subsequently.
• Use G.C. to find the probabilities for each value of X and look for the one with highest probability.
• The mode is usually near the expectation.

2. Additive properties of Poisson Random variable

• If X~Po(a) and Y~Po(b) , where X and Y are independent, then

X+Y ~ Po (a+b)

After this lecture, you can attempt Q1,2,3,4 (except (v)) ,Q5, Q6, Q7.

Lecture 3:Binomial and Poisson distribution

In this lecture we focus on the following points:

1. What is the characteristics of a Poisson distribution?

2. What is the probability distribution function, expectation and variance of Poisson distribution?

What is the characteristics of a Poisson distribution?

• The rate of occurance is constant throughout the given time interval or given space.
• The event happens singly and randomly and it is rare event.
• The probability of two events happening at the same time is negligible.
• The events happening in different time intervals or space are independent.

Examples of Poisson distribution includes no. of defects, demand of a certain item, no. of telephone calls etc.

What is the probability distribution function, expectation and variance of Poisson distribution?

• The p.d.f. of Poisson distribution is \[P(X=r)=\frac{e^{-\lambda}\lambda^{r}}{r!}\] where r=0,1,2,3,... (up to infinity)
• Expectation and variance: \[E(X)=\lambda\] \[Var(X)=\lambda\]

You can attempt Q1 - 3 in the tutorial 14B.

Lecture 2: Binomial & Poisson Distribution

In this lecture, we learnt three concepts: expectation, variance and mode

Expectation is the mean (average) value of the random variable after a large number of trials.

Variance is about the spread of the data, i.e. the larger the variance, the more likely that you will get a outcome that is far away from the expectation.

Mode is the value that is most likely to occur.

To do a question in Binomial distribution, you have to take note of the following:

• define the random variable first
• check that it does follow Binomial distribution
• find out what is the value of n (no. of trials) and p (probability of success)
• understand what the question is asking about and use G.C. to solve the problem

You can attempt all the questions in tutorial 14A after this lecture.

For enrichment: \[E(X)=\sum_{k=1}^{n}P(X=X_{k})X_{k}\]

\[Var(X)=\frac{1}{n}\sum_{k=1}^{n}P(X=X_{k})(X_{k}-E(X))^{2}\]

Challenging question: For a game, you have a probability of 0.5 to win 2 dollars, 0.4 chances to win 3 dollars and 0.1 chances to win 10 dollars. If you need to pay 3.5 dollars to play the game, would you want to play?

Lecture 1: Binomial & Poisson Distribution

In today's lecture, you have to take note of the following points:

1. What is a random variable? What is probability distribution?

2. Under what condition will the random variable follow Binomial distribution?

3. What is the probability distribution function for Binomial distribution?

What is a random variable? What is probability distribution?

Two important characteristics of a random variables are reflected in its name. " Variable" tells us that it is a value that can change. "Random" tells us that the change of the value is random, i.e. there is no way to predict the value.

There are two types of random variables: discrete and continuous. If you can count the possible values taken by the random variable, it is discrete. Otherwise, it is continuous.

A random variable can take a certain set of possible values. For each of these value, there is a certain probability associated. The way that the probability is distributed among these possible values is called the probability distribution. It can be summarized in terms of a formula or in a table form, which is called the probability distribution function of the random variable.

Under what condition will the random variable follow Binomial distribution?

• There are n independent trials. (Independence means that the outcome of any trials will not affect the outcome of rest of the trials. )
• For each trial, there are two possible outcomes, one success and one failure.
• The probability of success is constant in each trial.

Remark: It is very important to know the condition for each type of distribution. It helps you to determine the right distribution to use when given a problem.

3. What is the probability distribution function for Binomial distribution?

There are two important values to take note of for each binomial distribution: the no. of trials, n, and the probability of success, p. If X~B(n, p)

\[P(X=r)=^{n}\textrm{C}_{r}p^{r}q^{n-r}\]

After this lecture, you can attempt Tutorial 14A Q1

Lecture 3: probability

In the last lecture on probability, there are two emphasis.

1. Mutually exclusive and independent events

2. Tree diagram

Mutually exclusive and independent events

It is important to differentiate these two types of events.

• Mutually exclusive events mean that they can never happen together.
• Independent events means that the happening of one event will not affect the probability of the other happening.
• Mutually exclusive events are NOT independent because if one event happens, the probability of the other happening drops to zero! (It does affect the probability of the other event happening.)
• It is very difficult to judge whether two events are independent based on your intuition. Please use the definition of independent events P( A intersect B) =P(A)P(B) to check whether two events are independent.

Tree diagram

• Tree diagrams are useful if

1. The event has a few stages

2. The outcome in one stage will affect the probability in the next stage.

By the end of the lecture, you can attempt tutorial 13.

Lecture 1&2: Probability

Probability is a study of random events and chances. The modern theory of probability is founded by two mathematicians, Pascal and Fermat during the seventeenth century and it was started by the study of the chance of winning in a gambler's game.

In these two lectures, you need to understand the first few concepts.

1. The meaning of the basic terms: experiment, outcome, sample space and event

2. Classical method of obtaining probability (Important concept!!!): the probability of an event equals to the no. of ways that the event can occur dividing the total number of possible outcomes.

Remark: Classical method is useful only if each outcome are equally likely to happen.

3. Venn Diagram: Learn how to draw the Venn diagram and derive the formula from the Venn diagram

4. Conditional probability (Extremely important!!!):

• Definition: P(A|B)=P(A intersect B)/P(B)
• To calculate conditional probability, please make use of the definition rather than depending on your intuition. Your intuition is not always correct!

Tips for questions on probability:

• identity and define the events first
• change all the statements in terms of probability
• summarize the information in the question before you start to do the question

You can attempt all the questions in tutorial 13 except Q3 and 4.

Probability: Interesting Puzzle

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. He repeats this many times, each time betting half the total money he has. After 2n plays he has won exactly n times. Has he more money, the same amount or less money than he started with?

Permutations and Combinations: Lecture 2

In this lecture, you learn about combinations.

1. Notation of combinations and what does it mean?

If you choose r objects from n distinct objects disregard of the order of choosing, the no. of combination is represented by \[_{}^{n}\textrm{C}_{r}\]

Do you realise that combination notation is also used in binomial expansion? Why is it so?


2. Relationship between permutation and combination

\[_{}^{n}\textrm{P}_{r}={}^{n}\textrm{C}_{r}\times r!\]

Permutation means that the order of choosing is important while for combination the order is not important. Hence, the no. of ways of permutation with r objects chosen from n distinct objects is equivalent of the no. of ways of choosing r objects disregard of the order multiply by the no. of ways to arrange the r objects in order.

After this lecture, you can do tutorial 12.

Permutations and Combinations: Lecture 1

In this lecture, we learn about the following concepts.

1. the addition and multiplication principle
2. permutations

• permutation with distinct objects
• permutation with identical objects
• permutation allowing repetition
• circular permutation

The addition and multiplication principle

• The addition principle is used in the case that you have a few different ways to complete a task.
• The multiplication principle is used in the case that you have different stages in a task and you have a few ways to complete each stage.

Permutation with distinct objects

• When you choose r objects from n distinct projects (the order of choosing matters), the no. of ways that you can do so is \[_{}^{n}\textrm{P}_{r}\]

In the another way, we can use the multiplication principle. To choose the first object, we have n choices , i.e. n no. of ways. To choose the second object, we are left with (n-1) choices, i.e. (n -1) ways. Hence, the no. of ways to choose r objects is n(n-1)(n-2)...(n-r+1)

• In the GC, you can find "nPr" in MATH -> PRB

Permutation with identical objects

• If we are to arrange r objects with r1 of type 1, r2 of type 2,... and rn of type n, where r1+r2+...rn=r, the no. of ways is given by r!/(r1!r2!...rn!)

• The rationale behind this formula:

r! is the no. of ways to arrange the objects if all of the objects are different. If some of them are identical, we actually overcount the no. of ways by a factor, which is the no. of ways we can arrange those identical objects.

Permutation allowing repetition

• If we choose r objects taken from n distinct objects in order, and each object can be used more than once, the no. of ways to do so is \[n^{r}\], since each time when we choose an object, we have n choices.

Circular permutation

• The no. of ways to arrange n distinct objects in a circle is (n-1)!
• The rationale behind the formula:

First of all, we can consider the permutation of n distinct objects, which is n!. However, when we arrange them in a circle, we found that some of the arrangement becomes identical. In fact, since we only care about the order in the clockwise manner and ignore the starting position, every identical arrange is repeated n times. Hence, in the end, the no. of different arrangement is n!/n, which is (n-1)!

Remark: circular permutation with identical objects do not have a general formula. You might want to think about the reason.

Differential Equation: Lecture 4

In today's lecture, we learnt more about the application of DE in real-life.

1. Radioactive decay (You need to understand the term "half life", which means the time taken for the material to distintegrate into half of its original amount. )

2. Newton's law of cooling (The rate of decreasing of a object's temperature is directly proportional to the temperature difference between the object and the surrounding. )

3. Change of volume with time.

For this type of questions, you can follow the steps below to solve the problem.

- Look for key phrases that gives you the differential equation, which is usually a statement on the "rate of increase/decrease".

- Form the DE and solve it to have a general solution. (A solution might have one or two unknown constants.)

-Look for other data given in the questions. Use these data to solve for the unknown constants in the previous step.

By now we have finished the chapter DE, you can attempt all the tutorial questions.

Differential Equation: Lecture 3

In lecture 3, you learnt about two main concepts.

1. What is the family of curves?

2. How to model a question using DE and solve the problem?

1. Family of curves

The general solution of a DE can be represented by a set of curves graphically ( one curve for each value of the arbitrary constant). This set of curve is called the family of solution curves of the DE.

A particular solution of the DE corresponds to one specific curve in the family.

When you are asked to sketch the family of curves, you usually choose C=-1, 0 and 1 unless otherwise specified by the question.

Please use G.C. to help you in the sketching!!!

2. Modelling using DE

DE can be used to model a lot of real-life problems including physical phenomenon, process, human behaviours and population growth.

To model a problem with DE, follow the steps below.

Step 1: look for phrases like " the rate of increasing/decreasing of ... is proportional to ..." or ".... is increasing/decreasing at the rate proportional to... "

These phrases give you the Differential Equation.

Step 2: Solve the DE, usually by direct integration. The solution usually has some unknown constants.

Step 3: Use the given data in the question to find out the unknown constants.

After lecture 3, you will be able to do the whole tut11b if you are good at modelling. However you might wanna attempt Q1, 2, 3 , 6 first since they are similar to what we have gone through in the lecture.

Differential Equation: Lecture 2

In lecture 2, we learnt two main skills.

1. How to solve a second order DE in the form of \[\frac{\mathrm{d^2}y }{\mathrm{d} x^2}=f(x)\]

2. How to use substitution to solve a first order DE that cannot be solved by direct integration.


1. How to solve a second order DE

Step 1: integrate both sides once to get a first order DE (the first order DE obtained will have one arbitrary constant.)

Step 2: integrate again to get the general solution. (the general solution will have 2 arbitrary constant.)

2. How to use substitution to solve a first order DE that cannot be solved by direct integration.

Step 1: differentiate the given formula for substitution. (e.g. z=1/y)

Step 2: replace the dy/dx in the original DE by dz/dx

Step 3: observe whether there are still y in the DE, replace the y using substition or other methods.

The final DE should be in term of x and z and should be solvable by direct integration.

Step 4: solve the DE to obtain a relation between x and z.

Step 5: replace z by y using the substition given in the question.

After lecture 2, you can do the whole tutorial 11a.