Learn about joint probability density functions, marginal densities, and independence in the context of continuous random variables. Explore examples and concepts presented in "Introduction to Probability for Computing" by Harchol-Balter.
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Chapter 8 Continuous Random Variables: Joint distributions "Introduction to Probability for Computing", Harchol-Balter '24 1
Joint Densities Defn: The joint probability density function between continuous random variables ? and ? is a non-negative function ??,??,? , where ? ? ??,??,? ???? = ?{? ? ? & ? ? ?} ? ? and where ??,??,? ???? = 1 2 "Introduction to Probability for Computing", Harchol-Balter '24
Joint Densities Volume under the curve equals: ? ? ??,??,? ???? = ?{? ? ? & ? ? ?} ? ? 3 "Introduction to Probability for Computing", Harchol-Balter '24
Example Two-year-olds range in weight from 15 35 pounds. They range in height from 25 40 inches. ??,?(?, ) denotes the joint p.d.f. of weight and height. Q: What is the fraction of two-year-olds with weight > 30 pounds but height < 30 inches? A: =30 ?=30 ?= ??,??, ??? = 25 30 30 35??,??, ??? = Why are these the same? 4 "Introduction to Probability for Computing", Harchol-Balter '24
Marginal densities Defn: The marginal densities??(?) and ??(?) are defined as: ??(?) = ??,??,? ?? ??? = ??,??,? ?? Note that ??(?) and ??(?) are densities and not probabilities. Q: If ??,?(?, ) is the joint p.d.f. of weight and height in two-year-olds, what is the fraction of two-year-olds whose height is exactly 30 inches? ?= A: This is a zero-probability event! ??,??,30 ?? = ??(30) ?= 5 "Introduction to Probability for Computing", Harchol-Balter '24
Independence Defn: Continuous random variables ? and ?are i??????????, written ? ?, if: ??,??,? = ??? ??? ?,? Let s consider some joint p.d.f.s to determine whether ? and ? are independent. 6 "Introduction to Probability for Computing", Harchol-Balter '24
Example ? + ? if 0 ?,? 1 ??,?(?,?) = 0 otherwise Q: (a) What is ?[?]? (b) Is ? ?? A: part(a) 1 ? + ? ?? = ? +1 ??(?) = ??,??,? ?? = 2 0 1 ? +1 7 12 ? ? = ??? ??? = ? ?? = 2 0 7 "Introduction to Probability for Computing", Harchol-Balter '24
Example ? + ? if 0 ?,? 1 ??,?(?,?) = 0 otherwise Q: (a) What is ?[?]? (b) Is ? ?? A: part(b) ??,??,? ?? = ? +1 ??(?) = 2 ??,??,? ?? = ? +1 ??(?) = 2 Clearly, ??,??,? ??? ??? 8 "Introduction to Probability for Computing", Harchol-Balter '24
Example 4?? if 0 ?,? 1 ??,?(?,?) = 0 otherwise Q: Is ? ?? 1 A: ??(?) = 4?? ?? = 2? 0 1 ??(?) = 4?? ?? = 2? 0 Clearly, ??,??,? = ??? ??? 9 "Introduction to Probability for Computing", Harchol-Balter '24
Example: Which Exponential happens first? The time until server 1 crashes is ? ??? ? The time until server 2 crashes is ? ???(?) ? ??? ? ? ???(?) Q: What is the probability that server 1 crashes before server 2? Assume ? ?. A: ? ? < ? = ??,??,? ???? ?=0 ?=? = ??? ??? ???? ?=0 ?=? What happens when ? = ? ? ? ?? ?? ?? ?????? = = ? + ? ?=0 ?=? 10 "Introduction to Probability for Computing", Harchol-Balter '24
Conditional p.d.f. and Bayes Law Defn: Given two continuous random variables, ? and ?, we define the conditional p.d.f. of r.v. ? given event ? = ? as: =??|?=?? ??(?) ??(?) =??|?=?? ??(?) ???,?(?,?)?? ??|?=?? =??,?(?,?) ??(?) This is the definition of the conditional p.d.f., where we re conditioning on a zero-probability event ? = ? Here we ve used the same definition but this time applied it to conditioning on ? = ?,resulting in a Bayes Law for two continuous r.v.s Here we ve simply expanded out ??(?) ??|?=?? ?? = 1 Observe that the conditional p.d.f. is still a proper p.d.f., i.e., 11 "Introduction to Probability for Computing", Harchol-Balter '24
Law of Total Probability Generalized Recall the Law of Total Probability, repeated below: Theorem: Let ? be an event and ? be a continuous r.v. Then we can compute ?{?} by conditioning on the value of ? as follows: ? ? = ??? ? ?? = ? ? ? = ?} ??? ?? Using the definition for the conditional p.d.f. from the prior slide, we can similarly express ??? by conditioning on the value of ?: Theorem: Let ? and ? be continuous random variables. Then, from the definition of the conditional p.d.f., we have: ??? = ??,??,? ?? = ??|?=?? ??? ?? ? ? 12 "Introduction to Probability for Computing", Harchol-Balter '24
Example: Which Exponential happens first? The time until server 1 crashes is ? ??? ? The time until server 2 crashes is ? ???(?) ? ??? ? ? ???(?) Q: What is the probability that server 1 crashes before server 2? Assume ? ?. A: ? ? < ? = ? ? < ? ? = ?} ??? ?? 0 ? ? > ? ? = ?} ?? ???? = 0 ?{? > ?} ?? ???? = 0 Where did we use independence? ? ? ?? ?? ???? = = ? + ? 0 13 "Introduction to Probability for Computing", Harchol-Balter '24
From Midterm 2020 Random variables ? and ? are NOT independent. Their joint density is: ??,??,? where 0 ?,? All your answers should be in terms of ??,??,? or prior results. a. Write an expression for ??(?) b. Write an expression for ??|?=5(?) c. Write an expression for ? ? + ? < 10 ? = 5} d. Write an expression for ??|?<6(?) e. Write an expression for ??|?<6(?) 14 "Introduction to Probability for Computing", Harchol-Balter '24
From Midterm 2020 Random variables ? and ? are NOT independent. Their joint density is: ??,??,? where 0 ?,? Q: What is the mass of the blue event, relative to the world? All your answers should be in terms of ??,??,? or prior results. a. Write an expression for ??(?) ? The World ??? = ??,??,? ?? 0 ? ??? ? A: Blue event has zero probability 15 "Introduction to Probability for Computing", Harchol-Balter '24
From Midterm 2020 Random variables ? and ? are NOT independent. Their joint density is: ??,??,? where 0 ?,? Q: What is the mass of the blue event, relative to the world? All your answers should be in terms of ??,??,? or prior results. ? b. Write an expression for ??|?=5(?) ??|?=5? =??,?(?,5) ??(5) , 5 The World ? ? A: Blue event has zero probability 16 "Introduction to Probability for Computing", Harchol-Balter '24
From Midterm 2020 Random variables ? and ? are NOT independent. Their joint density is: ??,??,? where 0 ?,? Q: What is the mass of the blue event, relative to the world? All your answers should be in terms of ??,??,? or prior results. ? c. Write an expression for ? ? + ? < 10 ? = 5} 5 = ??|?=5? ?? , ? ? + ? < 10 ? = 5} , ?=0 5 The World 5??,?(?,5) ??(5) = ?? ? ?=0 5 A: Blue event has non-zero probability 17 "Introduction to Probability for Computing", Harchol-Balter '24
From Midterm 2020 Random variables ? and ? are NOT independent. Their joint density is: ??,??,? where 0 ?,? Q: What is the mass of the blue event, relative to the world? All your answers should be in terms of ??,??,? or prior results. ? d. Write an expression for ??|?<6(?) The World =??(? ? < 6) ?{? < 6} ??|?<6(?) , , ? 6 ??,??,? ?? = ?=0 ? 6 6 A: Blue event has zero probability ??? ?? ?=0 18 "Introduction to Probability for Computing", Harchol-Balter '24
From Midterm 2020 Random variables ? and ? are NOT independent. Their joint density is: ??,??,? where 0 ?,? All your answers should be in terms of ??,??,? or prior results. Q: What is the mass of the blue event, relative to the world? ? e. Write an expression for ??|?<6(?) 6 =??(? ? < 6) ?{? < 6} ??|?<6(?) , The World ? ??(?) ?{? < 6} if ? < 6 otherwise ? = 0 A: Blue event has zero probability 19 "Introduction to Probability for Computing", Harchol-Balter '24
Expectation with multiple r.v.s Defn: Let ? and ? be continuous random variables with joint p.d.f. ??,??,? . Then, for any function ? ?,? , we have ? ? ?,? = ? ?,? ??,??,? ???? 20 "Introduction to Probability for Computing", Harchol-Balter '24
Conditional expectation with multiple RVs Recall Defn: For a continuous r.v. ? and an event ?, where ? ? > 0,the conditional expectation of ? given ? is: ? ? ? = ? ??|?? ?? ? Defn: For continuous r.v.s ? and ? ? ??,?(?,?) ??(?) ? ? ? = ? = ? ??|?=?? ?? = ?? ? ? Theorem: We can derive ?[?] by conditioning on the value of continuous r.v. ? : ? ? = ? ? ? = ?] ??? ?? ? 21 "Introduction to Probability for Computing", Harchol-Balter '24
Example Two-year-olds range in weight from 15 35 pounds. They range in height from 25 40 inches. Q: My 2-year old is 30 inches tall. What is their expected weight? The World ? A: 35 ? ? ? = 30] = ? ??|?=30? ?? ?=15 35 ? ??,?(?,30) ??(30) = ?? ?=15 Why are these the same? ? 30 22 "Introduction to Probability for Computing", Harchol-Balter '24
Example Two-year-olds range in weight from 15 35 pounds. They range in height from 25 40 inches. Q: What fraction of 2-year olds with height 30 inches have weight < 25 pounds? The World ? A: 25 ? ? < 25 ? = 30} = ??|?=30? ?? ?=15 25 25 ??,?(?,30) ??(30) = ?? ? ?=15 30 23 "Introduction to Probability for Computing", Harchol-Balter '24
Example: Hand-in Time versus Grade ? = number of days early that homework is submitted: 0 ? 2 ? = grade on homework (as a percentage): 0 ? 1 Joint density function: 0 ? 2, 0 ? 1: 9 10tg2+1 ??,??,? = 5 Q: What is the probability that a random student gets a grade above 50%? A: 2 2 9 10tg2+1 ?? =9 5?2+2 ??(?) = ??,??,? ?? = 5 5 0 0 1 1 ? ? >1 9 5g2+2 = ??? ?? = ?? = 0.725 2 5 0.5 0.5 24 "Introduction to Probability for Computing", Harchol-Balter '24
Example: Hand-in Time versus Grade ? = number of days early that homework is submitted: 0 ? 2 ? = grade on homework (as a percentage): 0 ? 1 Q: Given that a student submitted less than a day before the deadline, does the probability of getting a grade >50% go down? ?=1 ?=0 ?=1??,??,? ?? ?? ?=0.5 ? ? > 0.5 ? < 1 =?{? > 0.5 & ? < 1} = A: ?=1??? ?? ?{? < 1} ?=0 9 10tg2+1 3 10? +1 ?=1 ?=0 ?=1 5?? ?? ?=0.5 = = 0.66 ?=1 5?? 9 10tg2+1 ?=0 ??,??,? = 5 1 1 9 10tg2+1 3 10? +1 ??? = ??,??,? ?? = ?? = 5 5 0 0 25 "Introduction to Probability for Computing", Harchol-Balter '24
Example: Hand-in Time versus Grade ? = number of days early that homework is submitted: 0 ? 2 ? = grade on homework (as a percentage): 0 ? 1 Q: A student submits at ? = 0, i.e., exactly when the homework is due. What is their expected grade? 1 1 ? ??,?(?,0) ??(0) ? ? ? = 0] = ? ??|?=0? ?? = ?? A: ?=0 ?=0 1 5 1 5 1 = 0.5 = ? ?? 9 10tg2+1 ?=0 ??,??,? = 5 1 1 9 10tg2+1 3 10? +1 ??? = ??,??,? ?? = ?? = 5 5 0 0 26 "Introduction to Probability for Computing", Harchol-Balter '24
Example: Hand-in Time versus Grade ? = number of days early that homework is submitted: 0 ? 2 ? = grade on homework (as a percentage): 0 ? 1 Q: By contrast, what is the expected grade of a student who submits > 1 day early? 1 1 ? ??(? 1 < ? < 2) ?{1 < ? < 2} = ?? ? ? 1 < ? < 2] = ? ??|1<?<2? ?? A: ?=0 ?=0 9 10tg2+1 2??,??,? ?? 2??? ?? 1 ? 1 ??,??,? = 5 = ?? 1 ?=0 1 3 10? +1 9 10tg2+1 3 10? +1 ??? = ??,??,? ?? = 2 5?? 5 1 1 0 = ? ?? = 0.673 2 5?? ?=0 1 27 "Introduction to Probability for Computing", Harchol-Balter '24
