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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 10

Question

Math 6010 • Linear Models 2004 • Question 10

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with \(P\{X_i=0\}=\theta \) and \(P\{X_i=1\}=1-\theta \) \((0<\theta <1)\). Find the numerical sequences \(a_n\) and \(b_n\) such that \[ Y_n=a_n\hat {X}_n(1-\hat {X}_n)-b_n \] converges in distribution to a nondegenerate random variables, where \[ \hat {X}_n=\frac {1}{n}\sum _{1\leq i\leq n}X_i. \]

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed random variables with \(P\{X_i=0\}=\theta\) and \(P\{X_i=1\}=1-\theta\) \((0<\theta<1)\). Find the numerical sequences \(a_n\) and \(b_n\) such that
\[
Y_n=a_n\hat{X}_n(1-\hat{X}_n)-b_n
\]
converges in distribution to a nondegenerate random variables, where
\[
\hat{X}_n=\frac{1}{n}\sum_{1\leq i\leq n}X_i.
\]

Topic Tags

Covariance and Independence Laws of Large Numbers Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits