Let X, Y be two Bernoulli random variables and denote by p = P[X = 1], q = P[Y = 1] and r = P[X = 1, Y = 1]. Let (X1, Y1), . . . , (Xn, Yn) be a sample of n i.i.d. copies of (X, Y ). Based on this sample, X and Y are independent, and r = pq. Define a=(summation (i=1 to n) (X i))/n and b=(summation (i=1 to n ) (X i))/n, c =((summation (i=1 to n)) (XiYi))/n. =>Show that the vector (a,b,c) is asymptotically normal and find the asymptotic covariance matrix.
Let X, Y be two Bernoulli random variables and denote by p = P[X = 1], q = P[Y = 1] and r = P[X = 1, Y = 1]. Let (X1, Y1), . . . , (Xn, Yn) be a sample of n i.i.d. copies of (X, Y ). Based on this sample, X and Y are independent, and r = pq. Define a=(summation (i=1 to n) (X i))/n and b=(summation (i=1 to n ) (X i))/n, c =((summation (i=1 to n)) (XiYi))/n. =>Show that the vector (a,b,c) is asymptotically normal and find the asymptotic covariance matrix.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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Let X, Y be two Bernoulli random variables and denote by p = P[X = 1], q = P[Y = 1] and r = P[X = 1, Y = 1].
Let (X1, Y1), . . . , (Xn, Yn) be a sample of n i.i.d. copies of (X, Y ). Based on this sample, X and Y are independent, and r = pq.
Define a=(summation (i=1 to n) (X i))/n and b=(summation (i=1 to n ) (X i))/n, c =((summation (i=1 to n)) (XiYi))/n.
=>Show that the
Please help me by showing that the vector (a,b,c) is asymptotically normal and find the asymptotic covariance matrix.
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