Chapter 1 ІМќ Discrete Probability Distribution: Binomial Distribution INSTITUT MATEMATIK K E J U R U T E R A A N U N I M A P

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

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

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) = nCxpx(1-p)n-x *Page 221 (text book) *in text book π is used instead of p Lecture 4 Abdull Halim Abdul

Example If X~B(4,0.1) then P(X=3) = 4C3(0.1)3(0.9)1 = 0.0036 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 4 Abdull Halim Abdul

Using excel will be discussed using a few examples. 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. Lecture 4 Abdull Halim Abdul

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. No. of cars passing a toll booth in one hour. No. defects in a square meter of fabric No. of network error experienced in a day. Lecture 4 Abdull Halim Abdul

Poisson distribution is characterised by the mean success, λ. And is denoted by Po(λ) Lecture 4 Abdull Halim Abdul

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

Example If X~ Po(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 4 Abdull Halim Abdul

Using excel will be discussed using a few examples. 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. Lecture 4 Abdull Halim Abdul

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 4 Abdull Halim Abdul

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 4 Abdull Halim Abdul

Normal distribution is characterised by its mean, μ and its std deviation, σ. And is denoted by N(μ , σ2) Lecture 4 Abdull Halim Abdul

If a random variable X is distributed normal with the mean, μ and its std deviation, σ then X ~ N(μ , σ2) The probability distribution function of X is *Page 250 (text book) Lecture 4 Abdull Halim Abdul

Then P(X = a) = 0, where a is any constant. If X ~ N(μ , σ2) 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 4 Abdull Halim Abdul

Using excel will be discussed using a few examples. 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. Lecture 4 Abdull Halim Abdul