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Unformatted text preview: STATISTICS 321 – Dr. Soma Roy Chapter 4: Commonly Used Distributions This chapter is divided into 2 sections: Section 1: Discrete Distributions, Section 2: Continuous Distributions 1 Some Discrete Distributions 1.1 The Bernoulli Distribution Consider an experiment where only one of two outcomes are possible – “success” and “failure.” Let the P(“success”) = p , and P(“failure”) = . Such a trial is called a Bernoulli trial. Example 1: Toss a fair coin – heads or tails. If “success”= heads, what is p ? For any Bernoulli trial, we define a random variable X as follows: Thus, the pmf for a Bernoulli random variable X is given by, Example 2: When circuit boards used in the manufacture of compact disc players are tested, the long–run percentage of defectives is 5%. A circuit board is chosen at random. Let X = 1 if the circuit board is defective, and X = 0 otherwise. What is the distribution of X ? Mean and Variance of a Bernoulli Random Variable : Using concepts learnt in Chapter 2 we can verify that μ = p and σ 2 = p (1 p ) 1 1.2 The Binomial Distribution The binomial experiment has the following structure: 1. The experiment consists of n trials. 2. Each trial can result in only one of two possible outcomes denoted by success (S) and failure (F) . 3. The trials are independent 4. The probability, p , of success is constant from trial to trial. Thus, a Binomial experiment is a collection of n Bernoulli experiments. The random variable X is the number of successes in the n trials with probability of success p at each trial. Let us work on what the pmf would look like. We may write, X has a binomial distribution with parameters n and p , denoted by X ∼ Bin( n,p ) • Note: It can be shown that each 0 ≤ p ( x ) ≤ 1 and ∑ all x p ( x ) = 1 . • Example – Give me a discount : A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all invoices, 10% receive the discount. In a company audit, 12 invoices are sampled at random. What is the probability that fewer than 4 of the sampled invoices receive the discount? 2 • Mean and Variance of a Binomial Random Variable : If X ∼ Bin ( n,p ), then Mean of X = E ( X ) = np and Variance of X = Var( X ) = npq, where q=1p. • Recall example – Give me a discount : In a random sample of 12 invoices, what is the average number of invoices that receive the discount? 1.3 The Poisson Distribution Let X = number of occurrences of an event in a given interval of time or space. Then, X is said to have a Poisson distribution with parameter λ if the pmf for X is given by p ( x ; λ ) = e x λ x x ! , x = 0 , 12 ,... • Note: It can be shown that each 0 ≤ p ( x ) ≤ 1 and ∑ all x p ( x ) = 1 ....
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 Normal Distribution, Lognormal Distribution

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