STPM 2015 Mathematics (T) Term 3 Assignment

Introduction

Binomial distribution may be approximated, under certain circumstances, by Poisson distribution, or normal distribution. One practical advantage is that the calculations for finding probabilities are much less tedious to perform.

Sample Question

Binomial distribution may be approximated, under certain circumstances, by Poisson distribution or normal distribution. One practical advantage is that the calculations for finding probabilities are much less tedious to perform.

1. Consider a random variable having binomial distribution B(n, p).

(a) Let p=0.05 and n=5, 10 and 20. Tabulate all the probabilities using binomial distribution, Poisson distribution and normal distribution. (Note: For n=5. manual calculations must be used)

(b) Repeat step 1(a) for p = 0.1 and 0.5.

2

(a) Illustrate graphically the probability distributions.

(b) Compare the probability distributions obtained and discuss your findings.

3. Investigate what happens when n increases.

P(X=4)=0 P(X=5)=0

Remark: sum of probability=0.9381

Remark. For normal distribution’s calculation above, the total probabilities

are 0.9381, so i will suggest you add another 0.0619 to the probability for x=0. So

that the total can be 1.00. Please ask your teacher and let me know.

Sample Graph

Graphs of binomial and Poisson distributions based on the above table. You should construct similar graph for normal distributions.

Graph-of-binomial n=10, p=0.05

Graph-of-binomial n=10, p=0.05

Sample Solution for n=10, p=0.05

For binomial distribution B(n,p) where n=10, p=0.05, below are the sample tables and graphs for the distributions. Please do the calculations yourself.

Sample solutions for others

For n=20, p=0.05

For n=5, p=0.1

For n=5, p=0.5

For n=10, p=0.1

For n=10, p=0.5

For n=20, p=0.1

For n=20, p=0.5

For non KK LEE students, please do yourself.

For KK LEE students, please do yourself and visit the students corner at kkleemaths.com for downloadable excel file of all the tables and graphs for reference. We will discuss in class soon.

Comparison for p=0.05 (different values of n)

The shapes of the Poisson distribution and the binomial distribution are similar for p=0.05.

The graphs are not symmetrical an skewed to the left. So normal approximation are not suitable in these cases. When n increases, the distributions for p=0.05 are also skewed to the left. When the n is more than 100, the binomial distributions are almost symmetrical. The shape of binomial distributions tends to symmetrical as the n increases after n=100.

Comparison for p=0.1 and 0.5 (different values of n)

Please do yourself. Construct the graphs and see and investigate.

** For p=0.5, as n increases, the shape of binomial distribution is always symmetrical.

** You can try to make conclusion about the conditions for normal approximations(np>5, nq>5) and Poisson approximations(small p).

I will list down all the requirements from all the school teachers that i received here. Please share. Cheers

** Do for p=0.9

20 Comments

1.

Boi boi on July 19, 2015 at 3:05 pm

Dear sir,for example n =20, do we need calculate untill p(x=20)..and how many significant number we should take?

Reply

o

KK LEE on July 20, 2015 at 11:53 pm

Hi. It depends on your teacher. For me, YES!

2.

cheng shen sing on July 28, 2015 at 6:56 pm

Sir, for the graph, you said do the calculation but what is that calculation you mean? Can you give me some hints about Q2b and 3

Reply

KK LEE on July 29, 2015 at 10:21 am

You need to show all the calculations(or workings) to obtains the probabilities.

cheng shen sing on August 1, 2015 at 10:46 pm

Not really understand. Can explain more?

KK LEE on August 2, 2015 at 11:08 pm

Similar to the calculation for n=5 and p=0.05. Please use the formula to find all the probabilities.

Question mark? on August 7, 2015 at 12:04 am

Sir, can I use smooth line graph instead of bar graph?

Reply

KK LEE on August 7, 2015 at 12:07 am

Vertical line

Afiqah on August 9, 2015 at 2:19 pm

Sir,how to compare the graph if I do 3 probabilities(binomial,poisson,normal) in one graph

KK LEE on August 9, 2015 at 11:54 pm

Compare their shapes

Wan on August 11, 2015 at 10:28 pm

sir,how to do Q2b and Q3? because i not really understand.

Reply

KK LEE on August 11, 2015 at 11:04 pm

Please refer the above

V on August 14, 2015 at 7:32 pm

Sir, for the normal distribution involving n=5 and p=0.05,since usually normal distributions involve P(Xx), using P(X=x) = P(x-0.5<X<x+0.5) would naturally mean using normal distribution as an approximate for binomial distribution..so maybe that is why it is off by 0.0619still, for this question, we can't do using normal distribution..so we have to use the continuity correction method, correct?

Reply

KK LEE on August 14, 2015 at 9:09 pm

Yes

V on August 14, 2015 at 9:41 pm

Oh, ok..thank you sir

tiya on August 15, 2015 at 11:49 am

sir, i dont quite understand the calculation for the normal distribution part. why is the x is between -0.5 and 0.5

Reply

KK LEE on August 16, 2015 at 12:03 am

Continuity correction.

Jakiro on August 15, 2015 at 8:25 pm

Sir, may i ask the purpose of adding another 0.0619 to the probability for x=0 (Normal distribution). That won’t be the correct value for P(X=0) isn’t it ?

Reply

KK LEE on August 16, 2015 at 12:04 am

What i mean is if your school teacher wants the addition, then you add the values. Else, leave it there.

Jakiro on August 16, 2015 at 9:11 am

Alright, thank you sir

老師講要 line graph 跟 bar chart together我連這個都做不出來~

第 2 a，我不懂我對不對，用 excel 打出來了，找 formula，poison 的是吗？我们只做 bar chart 罢了.... 请问第二题(a)的 parameter 要放什么号码？我真的完全不懂第二题怎样做 1 3 5 7 9 ... 25

ntro 是看題目要找些什么，（我也不太清楚有些什麽，只是確定，要有 normal distribution,bell shape curve）wikipedia 都可以找到了，課本也有一點。