Chapter04

# Chapter04 - Chapter 4 Commonly Used Distributions 4.1 The...

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Chapter 4. Commonly Used Distributions

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4.1 The Bernoulli Distribution Bernoulli trial: If there is an experiment that can result in one of two outcomes, and one outcome is defined as “success” and the other “failure,” then the probability of success can be denoted by p and the probability of failure is 1–p . Such a trial is called a Bernoulli trial. For the Bernoulli trial with random variable X : if the experiment results in success, then X = 1. Otherwise X = 0. p (0) = P ( X = 0) = 1 – p , p (1) = P ( X = 1) = p The random variable X of Bernoulli distribution with parameter p is denoted by “ X ~ Bernoulli( p ) .”
Mean and Variance of a Bernoulli Random Variable If X ~ Bernoulli( p ) , then the mean and variance of a Bernoulli random variable are computed as follows: μ X = ( 0 )( 1 – p ) + ( 1 )( p ) = p σ 2 X = ( 0 – p ) 2 ( 1 – p ) + ( 1 – p ) 2 ( p ) = p ( 1 – p )

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Example 4.1 A coin has probability 0.5 of landing heads when tossed. Let X = 1 if the coin comes up heads, and X = 0 of the coin comes up tails. What is the distribution of X ? Sol) The success probability, P ( X = 1), is equal to 0.5, when heads comes up. Therefore X ~ Bernoulli(0.5). A die has probability1 /6 of coming up 6 when rolled. Let X = 1 if the die comes up 6, and X = 0 otherwise. What is the distribution of X ? Sol) The success probability p = P (X = 1) = 1/6. Therefore X ~ Bernoulli(1/6).
Example 4.2 Ten percent of the components manufactured by a certain process are defective. A component is chosen at random. Let X = 1 if the component is defective, and X = 0 otherwise. What is the distribution of X ? Sol) The success probability p = P ( X = 1) = 0.1. Therefore X ~ Bernoulli(0.1). Find μ X and σ 2 X . Sol) μ X = p = 0. 1, σ 2 X = p ( 1 p ) = ( 0. 1) ( 0. 9) = 0. 09.

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4.2 The Binomial Distribution If a total of n Bernoulli trials are conducted, and The trials are independent Each trial has the same success probability p X is the number of successes in the n trials then X has the binomial distribution with parameter n and p , denoted “ X ~ Bin( n , p ) .” The sample size should be small: if the sample size is no more than 5% of the population, the binomial distribution may be used.
Example 4.3 A fair coin is tossed 10 times. Let X be the number of heads that appear. What is the distribution of X ? Sol) There are 10 independent Bernoulli trials with each success probability of p = 0.5. The random variable X is equal to the number of successes in the 10 trials. Therefore X ~ Bin(10, 0.5). A lot contains several thousand components, 10% of which are defective. Seven components are sampled from the lot. Let X represent the number of defective components in the sample. What is the distribution of X ? Sol) The sample size is much smaller than the population (less than

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Chapter04 - Chapter 4 Commonly Used Distributions 4.1 The...

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