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

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

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

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

Example If X~B(4,0.1) then P(X=3) = 4 C 3 (0.1) 3 (0.9) 1 = P(X=4) = 4 C 4 (0.1) 4 (0.9) 0 = P(X≥4) = P(X=3)+P(X=4) = 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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