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1. The joint p.d.f of the r.v.'s X and Y is given in the following table y\x 0 1 3 2 1/8 1/16 3/16 1/8 1 2 1/16 1/16 1/8 1/4 (a) Calculate P(X ≤ 2, Y > 1). (b) Find the d.f Fx,y of X, Y. (c) Find the marginal p.d.f.'s fx of X and fy of Y.

1. The joint p.d.f of the r.v.'s X and Y is given in the following table y\x 0 1 3 2 1/8 1/16 3/16 1/8 1 2 1/16 1/16 1/8 1/4 (a) Calculate P(X ≤ 2, Y > 1). (b) Find the d.f Fx,y of X, Y. (c) Find the marginal p.d.f.'s fx of X and fy of Y.

A First Course in Probability (10th Edition)
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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### Joint Probability Distribution Function The joint probability distribution function (p.d.f.) of the random variables (r.v.'s) \(X\) and \(Y\) is given in the following table: \[ \begin{array}{c|cccc} y \backslash x & 0 & 1 & 2 & 3 \\ \hline 1 & \frac{1}{8} & \frac{1}{16} & \frac{3}{16} & \frac{1}{8} \\ 2 & \frac{1}{16} & \frac{1}{16} & \frac{1}{8} & \frac{1}{4} \\ \end{array} \] ### Questions (a) Calculate \(P(X \leq 2, Y > 1)\). (b) Find the distribution function \(F_{X,Y}\) of \(X, Y\). (c) Find the marginal p.d.f.'s \(f_X\) of \(X\) and \(f_Y\) of \(Y\). (d) Calculate \(E(X)\), \(E(X^2)\), \(Var(X)\), \(E(Y)\), \(E(Y^2)\), \(Var(Y)\), \(E(XY)\), \(Cov(X,Y)\) and \(\rho(X,Y)\). (e) Calculate the conditional p.d.f \(f_{X|Y}(x|y)\), \(y = 1, 2\). (f) Calculate \(E(X|Y = y)\), \(y = 1, 2\) and calculate \(E(E(X|Y))\). (g) Compare \(E(X)\) and \(E(E(X|Y))\). (h) Calculate the moment generating function \(M_{X,Y}\). ### Explanation of Table - The table is a joint probability distribution that shows the probabilities corresponding to different combinations of values for the random variables \(X\) and \(Y\). - The rows represent different values of \(Y\) (1 and 2), while the columns represent the values of \(X\) (0, 1, 2, and 3). - Each cell contains the probability \(P(X = x, Y = y)\) for the specific combination of the random variables. For example, the probability that \(X = 0\) and \(Y = 1\
Transcribed Image Text:### Joint Probability Distribution Function The joint probability distribution function (p.d.f.) of the random variables (r.v.'s) \(X\) and \(Y\) is given in the following table: \[ \begin{array}{c|cccc} y \backslash x & 0 & 1 & 2 & 3 \\ \hline 1 & \frac{1}{8} & \frac{1}{16} & \frac{3}{16} & \frac{1}{8} \\ 2 & \frac{1}{16} & \frac{1}{16} & \frac{1}{8} & \frac{1}{4} \\ \end{array} \] ### Questions (a) Calculate \(P(X \leq 2, Y > 1)\). (b) Find the distribution function \(F_{X,Y}\) of \(X, Y\). (c) Find the marginal p.d.f.'s \(f_X\) of \(X\) and \(f_Y\) of \(Y\). (d) Calculate \(E(X)\), \(E(X^2)\), \(Var(X)\), \(E(Y)\), \(E(Y^2)\), \(Var(Y)\), \(E(XY)\), \(Cov(X,Y)\) and \(\rho(X,Y)\). (e) Calculate the conditional p.d.f \(f_{X|Y}(x|y)\), \(y = 1, 2\). (f) Calculate \(E(X|Y = y)\), \(y = 1, 2\) and calculate \(E(E(X|Y))\). (g) Compare \(E(X)\) and \(E(E(X|Y))\). (h) Calculate the moment generating function \(M_{X,Y}\). ### Explanation of Table - The table is a joint probability distribution that shows the probabilities corresponding to different combinations of values for the random variables \(X\) and \(Y\). - The rows represent different values of \(Y\) (1 and 2), while the columns represent the values of \(X\) (0, 1, 2, and 3). - Each cell contains the probability \(P(X = x, Y = y)\) for the specific combination of the random variables. For example, the probability that \(X = 0\) and \(Y = 1\
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