table:YX01200.080.070.060.0110.060.100.120.0520.050.060.090.0430.020.030.030.03Determine each of the following probabilities:(a)𝑃(? = 2)(b)𝑃(? ≥ 2)(c)𝑃(? ≤ 2 and ? ≤ 2)(d)𝑃(? = ?)(e)𝑃(? > ?)176. [DeGroot, P129, #3] Suppose that X and Y have a discrete joint distribution for which the joint p.f. isdefined as follows:?(?, ?) = {?|? + ?|if ? = −2, −1, 0, 1, 2 and ? = −2, −1, 0, 1, 20otherwiseDetermine(a) the value of the constantc;(b)𝑃(? = 0 and ? = −2)(c)𝑃(? = 1);(d)𝑃(|? − ?| ≤ 1)177. [DeGroot, P129, #1] Suppose that the joint p.d.f. of a pair of random variables(?, ?)is constant onthe rectangle where0 ≤ ? ≤ 2and0 ≤ ? ≤ 1, and suppose that the p.d.f. is 0 off of this rectangle.(a) Find the constant value of the p.d.f. on the rectangle.(b) Find𝑃(? ≥ ?)178. [DeGroot, P129, #4] Suppose that X and Y have a continuous joint distribution for which the jointp.d.f. is defined as follows:?(?, ?) = {??2if 0 ≤ ? ≤ 2 and 0 ≤ ? ≤ 10otherwiseDetermine(a) the value of the constantc;(b)𝑃(? + ? > 2)(c)𝑃(? < 1 2⁄ );(d)𝑃(? ≤ 1)(e)𝑃(? = 3?)179. [DeGroot, P129, #5] Suppose that the joint p.d.f. of two random variables X and Y is as follows:?(?, ?) = {?(?2+ ?)if 0 ≤ ? ≤ 1 − ?0otherwiseDetermine(a) the value of the constantc;(b)𝑃(0 ≤ ? ≤ 1 2⁄ )(c)𝑃(? ≤ ? + 1);(d)𝑃(? = ?2180. [DeGroot, P129, #6] Suppose that a point(?, ?)is chosen at random from the regionSin thexyplane containing all points(?, ?)such that? ≥ 0,? ≥ 0, and4? + ? ≤ 4. Determine the joint p.d.f.of X and Y .340.010.020.030.04;..;;.2;).-

20181. [DeGroot, P129, #7] Suppose that a point(?, ?)is to be chosen from the squareSin thexy-planecontaining all points(?, ?)such that0 ≤ ? ≤ 1and0 ≤ ? ≤ 1. Suppose that the probability that thechosen point will be the corner(0, 0)is 0.1, the probability that it will be the corner(1, 0)is 0.2, theprobability that it will be the corner(0, 1)is 0.4, and the probability that it will be the corner(1, 1)0.1. Suppose also that if the chosen point is not one of the four corners of the square, then it will be aninterior point of the square and will be chosen according to a constant p.d.f. over the interior of thesquare. Determine(a)𝑃(? ≤ 1 4⁄ )(b)𝑃(? + ? ≤ 1)182. [Strait, P251, #10] Suppose that X and Y are continuous random variables with joint probabilitydensity function given by?(?, ?) = {(9 4⁄ )?2?2if − 1 < ? < 1 and − 1 ≤ ? ≤ 10elsewhereFind:(a)𝑃(? + ? ≥ 1)(b)𝑃(? ≥ 0.5)183. [Strait, P252, #15] Let X and Y be discrete random variable with joint probability functionfdefinedby the following values:?(−1, 0) = .2,?(−1, 1) = .1,?(−1, 2) = .05,?(0, 0) = .05,?(0, 1) = .2?(0, 2) = .3,?(3, 0) = .02,?(3, 1) = .04, and?(3, 2) = .04(a) Find the marginal probability function of X.(b) Find the marginal probability function of Y.(c) Find𝑃(−1 ≤ ? ≤ 2),𝑃(−1 ≤ ? ≤ 2)184. [DeGroot, P140, #2] Suppose that X and Y have a discrete joint distribution for which the joint p.f. isdefined as follows:?(?, ?) = {130(? + ?)if ? = 0, 1, 2 and ? = 0, 1, 2, 30otherwise(a)Determine the marginal p.f.’s of X and Y.(b) Are X and Y independent?185. [DeGroot, P140, #3]Suppose that X and Y have a continuous joint distribution for which the jointp.d.f. is defined as follows:?(?, ?) = {32?2if 0 ≤ ? ≤ 2 and 0 ≤ ? ≤ 10otherwise(a)Determine the marginal p.d.f.’s of X and Y.(b) Are X and Y independent?is,..

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