Given X + Y has a normal distribution, prove that X and Y each have a...

Given X + Y has a normal distribution, prove that X and Y each have a normal distribution if X and Y are iid random variables

Expert Solution

orchestra answered 1 year ago

Suppose the random variables X and Y form a bivariate normal distribution. You are given that...

Suppose the random variables X and Y form a bivariate normal distribution. You are given that E[X] = 3, E[Y ] = −2, σX = 4, and σY = 3. Find the probability that X and Y are within 3 of each other under the following additional assumptions: (a) Corr(X, Y ) = 0 (b) Corr(X, Y ) = −0.6

Given f(x,y) = 2 ; 0< x ≤ y < 1 a. Prove that f(x,y) is...

Given f(x,y) = 2 ; 0< x ≤ y < 1 a. Prove that f(x,y) is a joint pdf. b. Find the correlation coefficient of X and Y.

Given below is a bivariate distribution for the random variables x and y. f(x, y) x...

Given below is a bivariate distribution for the random variables x and y. f(x, y) x y 0.3 50 80 0.2 30 50 0.5 40 60 (a) Compute the expected value and the variance for x and y. E(x) = E(y) = Var(x) = Var(y) = (b) Develop a probability distribution for x + y. x + y f(x + y) 130 80 100 (c) Using the result of part (b), compute E(x + y) and Var(x + y). E(x...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y < y^(−1) then x < −1.

Suppose the joint probability distribution of X and Y is given by the following table. Y=>3...

Suppose the joint probability distribution of X and Y is given by the following table. Y=>3 6 9 X 1 0.2 0.2 0 2 0.2 0 0.2 3 0 0.1 0.1 The table entries represent the probabilities. Hence the outcome [X=1,Y=6] has probability 0.2. a) Compute E(X), E(X2), E(Y), and E(XY). (For all answers show your work.) b) Compute E[Y | X = 1], E[Y | X = 2], and E[Y | X = 3]. c) In this case, E[Y...

2. Prove the following properties. (b) Prove that x + ¯ xy = x + y.

2. Prove the following properties.(b) Prove that x + ¯ xy = x + y.3. Consider the following Boolean function: F = x¯ y + xy¯ z + xyz(a) Draw a circuit diagram to obtain the output F. (b) Use the Boolean algebra theorems to simplify the output function F into the minimum number of input literals.

X and Y are discrete random variables with joint distribution given below Y 1 Y 0...

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 distribution of X and verify your answers to part (c).

Let X and Y have joint discrete distribution p(x, y) = 3 20 (.5 x )...

Let X and Y have joint discrete distribution p(x, y) = 3 20 (.5 x ) (.7 y ), x = 0, 1, 2, . . . , and y = 0, 1, 2, . . .. Find the marginal probability function P(X = x). [hint: for a geometric series X∞ n=0 arn with −1 < r < 1, r 6= 0, then X∞ n=0 arn = a 1 − r ]

prove Xbar is a complete sufficient statistic in a normal distribution

for each of the five functions f1-(x,1/y) ,f2-(0,y) ,f3 (x+2y,y) ,f4-(x-2,y+1) ,f5 (x,y^3-y) prove or disprove...

for each of the five functions f1-(x,1/y) ,f2-(0,y) ,f3 (x+2y,y) ,f4-(x-2,y+1) ,f5 (x,y^3-y) prove or disprove that fn is distance-preserving

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