Chap_3_Special_distributions.pdf - Chapter 3 Some special...

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Chapter 3: Some special distributions Phan Thi Khanh Van E-mail: [email protected] June 6, 2020 (Phan Thi Khanh Van) Chap 3: Some special distributions June 6, 2020 1 / 79
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Contents I 1 Bernoulli distribution 2 Binomial distribution 3 Geometric distributions 4 Hypergeometric distribution 5 Negative binomial distributions 6 Poisson distributions. Poisson processes 7 Uniform distributions. 8 Normal distributions. Central limit theorems 9 Log-normal distributions 10 Exponential distributions 11 Erlang distributions and Gamma distributions 12 Weibull distributions 13 Beta distributions (Phan Thi Khanh Van) Chap 3: Some special distributions June 6, 2020 2 / 79
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Bernoulli distribution 1 Flip a coin. X is the number of heads. P ( X = 1) = 0 . 5 = P ( X = 0) . 2 A worn machine tool produces 1% defective parts. Let X = number of defective part produced in one trial. P ( X = 1) = 0 . 01 , P ( X = 0) = 0 . 99 3 Each sample of air has a 10% chance of containing a particular rare molecule. Let X = the number of air samples that contain the rare molecule when the next sample is analyzed. P ( X = 1) = 0 . 1 . 4 Of all bits transmitted through a digital transmission channel, 10% are received in error. Let X = the number of bits in error when the next bit is transmitted. P ( X = 1) = 0 . 1 , P ( X = 0) = 0 . 9 . 5 Of all patients suffering a particular illness, 35% experience improvement from a particular medication. A patient is administered the medication, let X = the number of patients who experience improvement. P ( X = 1) = 0 . 35 , P ( X = 0) = 0 . 75 . (Phan Thi Khanh Van) Chap 3: Some special distributions June 6, 2020 3 / 79
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Bernoulli distribution The random variable X is called a Bernoulli random variable if it takes the value 1 with probability p and the value 0 with probability q = 1 - p . Mean and Variance If X is Bernoulli random variable with parameter p , μ = E ( X ) = p , σ 2 = V ( X ) = p (1 - p ) . Example A worn machine tool produces 1% defective parts. Let X = number of defective part produced in one trial. P ( X = 1) = 0 . 01 , P ( X = 0) = 0 . 99 . E ( X ) = 0 . 01 V ( X ) = 0 . 01 × 0 . 99 = 0 . 099 . (Phan Thi Khanh Van) Chap 3: Some special distributions June 6, 2020 4 / 79
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Binomial distribution 1 Flip a coin 3 times. Let X = number of heads obtained. 2 A worn machine tool produces 1% defective parts. Let X = number of defective parts in the next 25 parts produced. 3 Each sample of air has a 10% chance of containing a particular rare molecule. Let X = the number of air samples that contain the rare molecule in the next 18 samples analyzed. 4 Of all bits transmitted through a digital transmission channel, 10% are received in error. Let X = the number of bits in error in the next 5 bits transmitted. 5 A multiple-choice test contains 10 questions, each with four choices, and you guess at each question. Let X = the number of questions answered correctly. 6 In the next 20 births at a hospital, let X = the number of female births. (Phan Thi Khanh Van) Chap 3: Some special distributions June 6, 2020 5 / 79
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Binomial distribution Flip 3 coins Flip 3 coins, consider the number of heads: x 0 1 2 3 P ( X = x ) 1 8 3 8 3 8 1 8 Camera flash tests The time to recharge the flash is tested in three cell-phone cameras. The probability that a camera passes the test is 0 . 8
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