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.