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University of Notre Dame *

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E2565

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Oct 30, 2023

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9/7/2023 [Last time] Random Variable (R.V.) A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment. Probability Mass Function (PMF) For a discrete random variable ? with possible values ? 1 , ? 2 , … , ? 𝑛 a probability mass function ?(𝒙) is a function such that: 1. ?(? 𝑖 ) ≥ 0 2. ∑ ?(? 𝑖 ) 𝑛 𝑖=1 = 1 → ?(? 𝑖 ) ≤ 1 3. ?(𝒙 𝒊 ) = 𝑷(𝑿 = 𝒙 𝒊 ) → 𝑫??𝒊?𝒊𝒕𝒊?? Cumulative Distribution Function (CDF) The cumulative distribution function ?(?) , is the probability that a random variable ? with a given probability distribution will be found at a value less than or equal to ? . ?(?) = 𝑃(? ≤ ?) = ∑ ?(? 𝑖 ) 𝑥 𝑖 ≤𝑥 For a discrete random variable ?, ?(?) satisfies the following properties: 1. ?(?) = 𝑃(? ≤ ?) = ∑ ?(? 𝑖 ) 𝑥 𝑖 ≤𝑥 2. 0 ≤ ?(?) ≤ 1 3. 𝐼? ? ≤ ?, ?ℎ?? ?(?) ≤ ?(?) [Note] Some properties of CDF Let ? (1) , ? (2) , … . , ? (𝑛) are all sorted possible values of ? . 1. ?(? (𝑛) ) = 1 2. ?(?) = 0, where ? < ? (1) 3. ?(? (𝑖) ) = 𝑃(? = ? (𝑖) ) = ?(? (𝑖) ) − ?(? (𝑖−1) ) 4. 𝑃(? > ? (𝑖) ) = 1 − ?(? (𝑖) ) Example 3-8: Sampling without Replacement A day’s production of 850 parts contains 50 defective parts. Two parts are selected at random without replacement. Let the random variable ? equal the number of defective parts in the sample. Find the cumulative distribution function of ? . <sol> Find PMF: ?(?) → ?? ℎ??? ?? ?𝑖?? 𝑃𝑀? ?(?) ?𝑖??? ? = 0,1,2

𝑃(? = 0) = ?(0) = 800 850 ∗ 799 849 = 0.8857479 𝑃(? = 1) = ?(1) = 800 850 ∗ 50 849 + 50 850 ⋅ 800 849 = 0.1108571 𝑃(? = 2) = ?(2) = 50 850 ∗ 49 849 = 0.003394998 ?(0) = 𝑃(? ≤ 0) = 𝑃(? = 0) = ?(0) = 0.8857479 ?(1) = 𝑃(? ≤ 1) = ?(0) + ?(1) = 0.8857479 + 0.1108571 = 0.996605 ?(2) = 𝑃(? ≤ 2) = 1 = ?(0) + ?(1) + ?(2) = ?(1) + ?(2) = 0.996605 + 0.003394998 = 1 ?(?) = { 0, 𝑖? ? < 0 0.885748, 𝑖? 0 ≤ ? < 1 0.996605, 𝑖? 1 ≤ ? < 2 1, 𝑖? ? ≥ 2 Example: Roll two fair dice Let ? be the random number defined by the sum of the two dice. Find PMF, CDF. ? ?(?) ?(?) <2 0 0 2 1/36 1/36 3 2/36 3/36 4 3/36 6/36 5 4/36 10/36 6 5/36 15/36 7 6/36 21/36 8 5/36 26/36 9 4/36 30/36 10 3/36 33/36 11 2/36 35/36 12 1/36 1 >12 0 1

Questions: 1. What is the probability of getting a total of 7 or 11? 2. What is the probability of getting a sum smaller than 7? 3. What is the probability of getting a sum greater than 5? <sol> 1. 𝑃(? = 7) + 𝑃(? = 11) = ?(7) + ?(11) = 8 36 2. 𝑃(? < 7) = 𝑃(? ≤ 6) = ?(6) = 15 36 3. 𝑃(? > 5) = 1 − 𝑃(? ≤ 5) = 1 − ?(5) = 26 36 Expected Value of a Function of a Discrete Random Variable: If X is a discrete random variable with probability mass function ?(?) , ?[ℎ(?)] = ∑ ℎ(?) ⋅ ?(?) 𝑥 ℎ(?) = ? Mean, Variance, & Standard Deviation: The mean or expected value of the discrete variable ?, denoted as 𝜇 or ?(?) , is 𝜇 = ?(?) = ∑ ? ⋅ ?(?) 𝑥 The variance of ? , denoted as 𝜎 2 or ?(?) , is 𝜎 2 = ?(?) = ?[(? − 𝜇) 2 ] = ∑(? − 𝜇) 2 ⋅ ?(?) 𝑥 = [∑ ? 2 ⋅ ?(?) 𝑥 ] − 𝜇 2 <pf> ?[(? − 𝜇) 2 ] = ∑(? − 𝜇) 2 ⋅ ?(?) 𝑥 = ∑(? 2 − 2?𝜇 + 𝜇 2 ) ⋅ ?(?) 𝑥 = ∑ ? 2 ⋅ ?(?) 𝑥 − 2𝜇 ∑ ? ⋅ ?(?) 𝑥 + 𝜇 2 ⋅ ∑ ?(?) 𝑥 = [∑ ? 2 ⋅ ?(?) 𝑥 ] − 𝜇 2 = ?[? 2 ] − (?[?]) 2 The standard deviation of ? is 𝜎 = √𝜎 2 .

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Ex: Roll a dice or a die once. Let ? be the random number defined by the number of the die. 𝜇 = ?(?) = ∑ ? ⋅ ?(?) 6 𝑥=1 = 1 ⋅ 1 6 + 2 ⋅ 1 6 + ⋯ + 6 ⋅ 1 6 = 21 6 = 7 2 𝜎 2 = ∑ ? 2 ⋅ ?(?) 6 𝑥=1 − 𝜇 2 = 1+4+9+16+25+36 6 − ( 7 2 ) 2 = 91 6 − 49 4 = 182−147 12 = 35 12 𝜎 2 = ∑ (? − 𝜇) 2 ⋅ ?(?) = (1− 7 2 ) 2 +(2− 7 2 ) 2 +(3− 7 2 ) 2 +(4− 7 2 ) 2 +(5− 7 2 ) 2 +(6− 7 2 ) 2 6 6 𝑥=1 = (− 5 2 ) 2 +(− 3 2 ) 2 +(− 1 2 ) 2 +(− 1 2 ) 2 +(− 3 2 ) 2 (− 5 2 ) 2 6 = 1+9+25 12 = 35 12 𝜎 = √ 35 12 Ex: Roll a die once. Let ? be the random number defined by the number of the dice. Find ?[? 3 ] ?[ℎ(?)] = ∑ ℎ(?) ⋅ ?(?) 𝑥 and ℎ(?) = ? 3 ∴ ?[? 3 ] = ∑ ? 3 ⋅ ?(?) 6 𝑥=1 = 1+8+27+64+125+216 6 = 441 6 = 73.5 Important Formulae: 1. ?(𝐶) = 𝐶, ?ℎ??? 𝐶 𝑖? ? ???????? 2. ?[𝐶 ⋅ ℎ(?)] = 𝐶 ⋅ ?[ℎ(?)] 3. ?[ℎ 1 (?) + ℎ 2 (?) + ⋯ + ℎ 𝑘 (?)] = ?[ℎ 1 (?)] + ?[ℎ 2 (?)] + ⋯ ?[ℎ 𝑘 (?)] Expected value and Variance of a linear function of a R.V. ?: Let the linear function ℎ(?) = ?? + ? . Then, 1. ?(?? + ?) = ? ⋅ ?(?) + ? 2. ?(?? + ?) = ? 2 ⋅ ?(?) <pf> By definition: 𝜎 2 = ?(?) = ?[(? − 𝜇) 2 ] ??? 𝜇 = ?(?) ?(?? + ?) = ?{[?? + ? − ?(?? + ?)] 2 } = ?{[?? + ? − ??(?) − ?] 2 } = ?{[?(? − ?(?))] 2 } = ? [? 2 (? − ?(?)) 2 ] = ? 2 ⋅ ? [(? − ?(?)) 2 ] = ? 2 ⋅ ?(?)