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Section 08 𝑓 𝑥, 𝑦 = 𝑃 𝑋 = 𝑥 ∩ 𝑌 = 𝑦 defined over a two-dimensional region Discrete: 𝑦𝑓 𝑥, 𝑦 = 1 Continuous: 𝑥 ∞ 𝑓 −∞ 𝑥, 𝑦 𝑑𝑦 𝑑𝑥 = 1 X and Y may be independent or dependent 𝐹 𝑥, 𝑦 = 𝑃[ 𝑋 ≤ 𝑥 ∩ 𝑌 ≤ 𝑦 ] Discrete: 𝑥 𝑠=−∞ 𝐹 𝑥, 𝑦 = 𝑦 𝑡=−∞ 𝑓(𝑠, 𝑡) Continuous: 𝐹 𝑥, 𝑦 = 𝜕2 𝐹 𝜕𝑥𝜕𝑦 𝑥 𝑦 𝑓 −∞ −∞ 𝑠, 𝑡 𝑑𝑡 𝑑𝑠 𝑥, 𝑦 = 𝑓(𝑥, 𝑦) If ℎ(𝑥, 𝑦) is a function of two variables, and X and Y are jointly distributed random variables, the expected value of ℎ(𝑥, 𝑦) is 𝐸 ℎ 𝑥, 𝑦 𝐸 ℎ 𝑥, 𝑦 = = continuous ℎ 𝑥, 𝑦 ∗ 𝑓(𝑥, 𝑦) for discrete ∞ ℎ −∞ 𝑥, 𝑦 ∗ 𝑓 𝑥, 𝑦 𝑑𝑦 𝑑𝑥 for If X and Y have joint probability distribution f(x,y) then the marginal distribution of X is 𝑓𝑋 𝑥 = 𝑓𝑋 𝑥 = 𝑦 𝑓(𝑥, 𝑦) for discrete ∞ 𝑓 𝑥, 𝑦 𝑑𝑦 for continuous −∞ Think of it as “adding up” the probability for all points where X is a certain value to get the overall probability that X=x. Random variables X and Y are independent if: The probability space is rectangular AND 𝑓 𝑥, 𝑦 = 𝑓𝑋 𝑥 ∗ 𝑓𝑌 (𝑦) The second condition can also be determined via 𝐹 𝑥, 𝑦 = 𝐹𝑋 𝑥 ∗ 𝐹𝑌 (𝑦) Recall that 𝑃 𝐴 𝐵 = 𝑃(𝐴∩𝐵) 𝑃(𝐵) The same concept is used to find conditional distributions 𝑃 𝑋 𝑌 = 𝑦 = 𝑓𝑋|𝑌 (𝑥|𝑌 = 𝑦) = 𝑓(𝑥,𝑦) 𝑓𝑌 (𝑦) Expectation: 𝐸 𝑌𝑋=𝑥 = 𝑦 ∗ 𝑓𝑌|𝑋 𝑦 𝑋 = 𝑥 𝑑𝑦 Variance: 𝑉𝑎𝑟 𝑌 𝑋 = 𝑥 = 𝐸 𝑌 2 𝑋 = 𝑥 − 𝐸 𝑌 𝑋 = 𝑥 2 = 𝑦 2 ∗ 𝑓𝑌|𝑋 𝑦 𝑋 = 𝑥 𝑑𝑦 − [ 𝑦 ∗ 𝑓𝑌|𝑋 𝑦 𝑋 = 𝑥 𝑑𝑦]2 𝐶𝑜𝑣 𝑋, 𝑌 = 𝐸 𝑋𝑌 − 𝐸 𝑋 ∗ 𝐸(𝑌) If covariance is zero, X and Y are independent Covariance is used to find the variance of the sum of X and Y 𝑉𝑎𝑟 𝑎𝑋 + 𝑏𝑌 = 𝑎2 𝑉𝑎𝑟 𝑋 + 𝑏 2 𝑉𝑎𝑟 𝑌 + 2𝑎𝑏𝐶𝑜𝑣(𝑋, 𝑌) Coefficient of correlation: 𝜌 𝑋, 𝑌 = 𝜌𝑋,𝑌 = 𝐶𝑜𝑣(𝑋,𝑌) 𝜎𝑋 𝜎𝑌 A device runs until either of two components fails, at which point the device stops running. The joint density function of the lifetimes of the two components, both measured in hours, is f(x, y) = (x+y)/8, for 0<x<2 and 0<y<2. Calculate the probability that the device fails during its first hour of operation An insurance company insures a large number of drivers. Let X be the random variable representing the company’s losses under collision insurance, and let Y represent the company’s losses under liability insurance. X and Y have joint density function 2𝑥 + 2 − 𝑦 for 0 < 𝑥 < 1 and 0 < 𝑦 < 2 𝑓 𝑥, 𝑦 = 4 0 otherwise What is the probability that the total loss is at least 1? A joint density function is given by f(x,y) = kx, 0<x<1 & 0<y<1 Where k is a constant. Calculate Cov(X,Y). Let X and Y be continuous random variables with joint density function f(x, y) = (8/3)xy, 0 < x < 1, x < y < 2x Calculate the covariance of X and Y. Let X and Y be continuous random variables with joint density function f(x,y) = 15y, x^2 <= y <= x Determine g, the marginal density function of Y, including its support. The distribution of Y, given X, is uniform on the interval [0,X]. The marginal density of X is f(x) = 2x, 0<x<1 Determine the conditional variance of X, given Y = y. An actuary analyzes a company’s annual personal auto claims, M, and annual commercial auto claims, N. The analysis reveals that Var(M) = 1600, Var(N) = 900, and the correlation between M and N is 0.64. Calculate Var(M+N). The joint probability density function of X and Y is given by f(x, y) = (x+y)/8 0<x<2 0<y<2 Calculate the variance of (X + Y)/2. A policyholder has probability 0.7 of having no claims, 0.2 of having exactly one claim, and 0.1 of having exactly two claims. Claim amounts are uniformly distributed on the interval [0,60] and are independent. The insurer covers 100% of each claim. Calculate the probability that the total benefit paid to the policyholder is 48 or less. X and Y have the joint cumulative distribution function F(x, y) = xy(x+y)/2,000,000 0<x<100; 0<y<100 Calculate Var(X). A city with borders forming a square with sides of length 1 has its city hall located at the origin when a rectangular coordinate system is imposed on the city so that two sides of the square are on the positive axes. The density function of the population is f(x,y) = 1.5(x^2 + y^2), 0<x,y<1 A resident of the city can travel to the city hall only along a route whose segments are parallel to the city borders. Calculate the expected value of the travel distance to the city hall of a randomly chosen resident of the city.