An urn contains four balls; two of the balls are numbered 1, and the other two are numbered 2. Twoballs are drawn without replacement. Let X be the smallest number of balls drawn and Y the largest.a) Find the joint density of X and Y.b) Find the marginal distribution of Y.c) Find Covariance(X, Y)

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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The magnitudes of Earthquake and Tsunami are given by the variables of m and n, respectively. Their joint occurrence probability is given by the joint probability mass function as: f (m, n) = A (m2 + n?) for the magnitudes of zero, one, two and three. a) Find the value of the constant A. b) Find the mean value of the joint occurrence function of these magnitudes. c) Determine the marginal distributions of these magnitudes. d) If an earthquake of magnitude 1 occurred, find the probability of having a tsunami of magnitude 2 and greater. e) Determine if the magnitude of earthquake and tsunami are independent random variables. Prove your opinion.
?please ans asap.
A machine generates rectangular tiles with randomly varying length (L) and width (W). Let the length follow L ~Gamma(100,10) distribution and the width follow W ~Gamma(50,10) distribution, and define the area of the tile as S = L × W, and its perimeter as Q = 2(L+W). For parts (a-c) below, assume the length and width are independent (i.e. LI W). (a) Show that the expected value of the area is E[S] = 50. (b) Show that the variance of the area is Var[S] = 75.5. (c) Use Chebyshev's inequality to find an upper bound on P(S > 75). For parts (d,e) below, assume the length and width are correlated with Corr(L, W) = -0.5. (d) Show that the expected value of the area is E[S] ~ 49.64645. (e) Find the mean and variance of the perimeter: E[Q] and Var[Q].
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b. The table below shows the joint probability mass function for the discrete random variables X and Y. Y = 1 Y = 2 X = 0 X = 1 X = 2 0.07 0.23 0.18 0.05 0.29 0.18 Calculate and interpret the covariance between variables X and Y i.e. Cov(X,Y).

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