1. Business Statistics (BQT 173) ІМ ќ INSTITUT MATEMATIK K E J U R U T E R A A N U N I M A P Discrete Probability Distribution: Binomial Distribution

2. Binomial Distribution Binomial distribution is the probability distribution of the number of successes in n trials. E.g. 1. No. of getting a head in tossing a coin 10 times. 2. No. of getting a six in tossing 7 dice. 3. No of missile hits the target. Lecture 4Abdull Halim Abdul2

3. Binomial distribution is characterised by 1. the number of trial, n and 2. the probability of success in each trial, p And is denoted by B(n,p) Lecture 4Abdull Halim Abdul3

4. If a random variable X is distributed binomial with the parameter n and p then X ~ B(n,p) The probability distribution of X is P(X=x) = n C x p x (1-p) n-x  *Page 221 (text book)  *in text book π is used instead of p Lecture 4Abdull Halim Abdul4

5. Example If X~B(4,0.1) then P(X=3) = 4 C 3 (0.1) 3 (0.9) 1 = 0.0036 P(X=4) = 4 C 4 (0.1) 4 (0.9) 0 = 0.0001 P(X≥4) = P(X=3)+P(X=4) = 0.0037 What if n is large? Calculation would be tedious. Solution… using cummulative binomial distribution table or statistical software or excel spreadsheet. Lecture 4Abdull Halim Abdul5

6. cummulative binomial distribution table already discussed at school. It will only be discussed in the tutorial. Using excel will be discussed using a few examples. examples Lecture 4Abdull Halim Abdul6

7. Business Statistics (BQT 173) ІМ ќ INSTITUT MATEMATIK K E J U R U T E R A A N U N I M A P Discrete Probability Distribution: Poisson Distribution

8. Poisson Distribution Poisson distribution is the probability distribution of the number of successes in a given space*.  *space can be dimensions, place or time or combination of them E.g. 1. No. of cars passing a toll booth in one hour. 2. No. defects in a square meter of fabric 3. No. of network error experienced in a day. Lecture 4Abdull Halim Abdul8

9. Poisson distribution is characterised by the mean success, λ. And is denoted by P o (λ) Lecture 4Abdull Halim Abdul9

10. If a random variable X is distributed Poisson with the parameter λ then X ~ P o (λ) The probability distribution of X is *Page 228 (text book) Lecture 4Abdull Halim Abdul10

11. Example If X~ P o (3) then P(X=2) = = 0.2240 P(X>2) = P(X=3)+P(X=4)+…+P(X=∞) = 1 – [P(X=0)+P(X=1)+P(X=2)] = 0.5768 To avoid tedious calculation it is easier to use cummulative Poisson distribution table or statistical software or excel spreadsheet. Lecture 4Abdull Halim Abdul11

12. cummulative binomial distribution table already discussed at school. It will only be discussed in the tutorial. Using excel will be discussed using a few examples. examples. Lecture 4Abdull Halim Abdul12

13. Business Statistics (BQT 173) ІМ ќ INSTITUT MATEMATIK K E J U R U T E R A A N U N I M A P Continuous Probability Distribution: Normal Distribution

14. Why Normal Distribution Numerous continuous variables have distribution closely resemble the normal distribution. The normal distribution can be used to approximate various discrete prob. dist. The normal distribution provides the basis for classical statistical inference. Lecture 4Abdull Halim Abdul14

15. Properties of Normal Distribution It is symmetrical with mean, median and mode are equal. It is bell shaped Its interquartile range is equal to 1.33 std deviations. It has an infinite range. Lecture 4Abdull Halim Abdul15

16. Normal distribution is characterised by its mean, μ and its std deviation, σ. And is denoted by N(μ, σ 2 ) Lecture 4Abdull Halim Abdul16

17. If a random variable X is distributed normal with the mean, μ and its std deviation, σ then X ~ N(μ, σ 2 ) The probability distribution function of X is *Page 250 (text book) Lecture 4Abdull Halim Abdul17

18. If X ~ N(μ, σ 2 ) Then P(X = a) = 0, where a is any constant. Previuosly it is very difficult to calculate the probability using the pdf. So all normal distribution is converted to std normal distribution, Z ~ N(0,1) for calculation. i.e Lecture 4Abdull Halim Abdul18

19. Calculation using std normal distribution table already discussed at school. It will only be discussed in the tutorial. Using excel will be discussed using a few examples. examples Lecture 4Abdull Halim Abdul19