4) (30 POINTS TOTAL) X an Y are discrete random variable; X has sample space 1,21...
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(4) (30 POINTS TOTAL) X and Y are discrete random variables; X has sample space 1,2} and Y has sample space 10, 1). Table 1 shows the joint distribution of (X, Y) TAbLe 1. Joint p.m.f. lC 4 (abcdef) Q搜索 (a) (5 points) Compute the marginal distribution of x and y, i.e. complete the following table 1 2 p(y) 1.3.4 p(x) (b) (5 points) Calculate the expectation of y, E[y] (c) (5 points) Calculate the conditional...
Problem The random variable X is exponential with parameter 1. Given the value r of X, the random variable Y is exponential with parameter equal to r (and mean 1/r) Note: Some useful integrals, for λ > 0: ar (a) Find the joint PDF of X and Y (b) Find the marginal PDF of Y (c) Find the conditional PDF of X, given that Y 2. (d) Find the conditional expectation of X, given that Y 2 (e) Find the...
Question 4: [10 Marks: 3, 3, 2, 2] X and Y are discrete random variables with joint distribution given below Y=-1 Y = 0 Y = 1 X= 1 0 1/4 0 X = 0 1/4 1/4 1/4 (a) Determine the conditional expectation E[Y|X = 1]. (b) Determine the conditional expectation E[X Y = y) for each value of y. (c) Determine the expected value of X using conditional expectation results form part (b) above. (d) Now obatin the marginal...
A discrete random variable ? has the sample space ?x = {1,2,3}, with given probabilities of ?x(1) = 0.3, ?x(2) = 0.4, and ?x(3) = 0.3. Compute the expectation ?[(? − ?)2]
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2. Let X and Y be discrete random variables with joint probability mass function X=1 X=5 Y=1 5a За Y=5 4a 8а a. What is the value of a? b. What is the joint probability distribution function (PDF) of X and Y? c. What is the marginal probability mass function of X? d. What is the expectation of X? e. What is the conditional probability mass function of X given Y = 1? f. Are X...
Two random variables, X and Y, have joint probability density function f ( x , y ) = { c , x < y < x + 1 , 0 < x < 1 0 , o t h e r w i s e Find c value. What's the conditional p.d.f of Y given X = x, i.e., f Y ∣ X = x ( y ) ? Don't forget the support of Y. Find the conditional expectation E [...
roblem 4 points A point A (X, Y, Z) in a three-dimensional Euclidean space R3 has the uniform joint distribution within the ball of radius 1 centered at the origin (OinR3.) Consider a random variable, T d (A, O), that is the distance from A to the origin. 1. Find the cumulative distribution function for T 2. Evaluate its expectation, E T] 3. Evaluate the variance, Var [T] .
MA2500/18 Section B (Answer THREE questions) 6. Let X and Y be jointly continuous random variables defined on the same prob- ability space, let fx.y denote their joint PDF, and let fx and fy respectively denote their marginal PDFs (a) Let z be a fixed value such that fx(x) >0. Write down expressions for 12] (i) the conditional PDF of Y given X = z, and (i) the conditional expectation of Y given X (b) State and prove the law...
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
1. Let X and Y be two discrete random variables each with the same the possible outcomes {1,2,3} a) Construct a bivariate probability mass function Px.y : {1,2,3} x {1,2,3} + R that satisfies the following properties propeties: (i) The expectation of X is E[X] = 2.1, (ii) The conditional expectation of Y given 2 = 3 is EY 2 = 3] = 1, (iii) The correlation between X and Y is slightly positive so that 0 < corr(X,Y) <...