Chapter04

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

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 4. Commonly Used Distributions
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ) .”
Background image of page 2
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 )
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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).
Background image of page 4
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.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 6
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
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 28

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

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online