Let \(\{X_n\}_{n \ge 1}\) be a sequence of independent random variables with identical Bernoulli distribution of parameter p. Find incorrect statements as n tends to infinity.
\(\frac{\frac{1}{n}\sum_{i=1}^{n} X_i - p}{\sqrt{p(1-p)/n}} \to 0\) with probability 1
Let \(\bar X_n = \frac{1}{n}\sum_{i=1}^{n} X_i\). By the weak law of large numbers \(\bar X_n \to p\) in probability, so the first statement is correct.
By the strong law, \(\bar X_n \to p\) almost surely, which is equivalent to \(P\left(\bigcup_{m\ge n}\{|\bar X_m - p| \gt \varepsilon\}\right) \to 0\) for every \(\varepsilon \gt 0\); so the second statement is also correct.
Since \(|\bar X_n - p| \le 1\) is bounded, convergence in probability implies \(L^1\) convergence; indeed \(E|\bar X_n - p| \le \sqrt{p(1-p)/n} \to 0\).
But \(\frac{\bar X_n - p}{\sqrt{p(1-p)/n}} = \frac{\sum X_i - np}{\sqrt{np(1-p)}}\) converges in distribution to N(0, 1) by the central limit theorem, so it cannot tend to 0 with probability 1 (by the law of the iterated logarithm its lim sup is in fact \(+\infty\) almost surely).
Check: for \(0 \lt p \lt 1\), \(P\left(\left|\frac{\sum X_i - np}{\sqrt{np(1-p)}}\right| \gt 1\right) \to 2(1-\Phi(1)) \approx 0.317\), not 0, so even convergence in probability to 0 fails.
Hence, \(\frac{\frac{1}{n}\sum_{i=1}^{n} X_i - p}{\sqrt{p(1-p)/n}} \to 0\) with probability 1.
Similar Questions
Let \(X\) be a connected metric space. Then which of the following is true ?
The series :
\(\frac{2}{1} + \frac{3}{4} + \frac{4}{9} + \ldots + \frac{n+1}{n^2} + \ldots\)
Let \(X, Y\) be metric spaces. Suppose \(f, g : X \to Y\) are continuous functions and \(D = \{x \mid f(x) = g(x)\}\). Then \(f \equiv g\) if \(D\) is :
Which of the following statements is not true in any metric space ?
Suppose that \(\{f_n\}\) is a sequence of functions defined by :
\(f_n(x) = \frac{x^n}{1 + x^n}, \ 0 \le x \le 1.\)
Then, which of the following statements is true ?
\(S_1 : f_n \to f \equiv 0\) uniformly on \([0, 1]\)
\(S_2 : \sup_{0 \le x \le \frac{1}{2}} f_n(x) \to F(x) \equiv 0\) on \([0, \frac{1}{2}]\)
\(S_3 : f_n \to f = 0\) uniformly on \([0, \frac{1}{2}]\)
Suppose that \(f\) is a function on \(\mathbb{R}^2\) defined by :
\(f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases}\)
Then, which of the following statements is/are true ?
\(S_1\) : \(f\) is continuous at \((0, 0)\).
\(S_2\) : Both partial derivatives exist at \((0, 0)\).
For any two sequences \(\{x_n\}\) and \(\{y_n\}\), it is known that,
\(\liminf x_n + \liminf y_n \le \liminf (x_n + y_n) \le \limsup (x_n + y_n) \le \limsup x_n + \limsup y_n \quad (*)\)
Suppose that \(x_n = \begin{cases} 0 & \text{if } n \text{ is even} \\ 1 & \text{if } n \text{ is odd} \end{cases}\) and \(y_n = 1 - x_n\), for all \(n \ge 1\).
Then, which of the following statements is/are true for \(\{x_n\}\) and \(\{y_n\}\) ?
\(S_1\) : strict inequalities hold in \((*)\) above.
\(S_2\) : \(\{x_n + y_n\}\) converges, while \(\{x_n\}\) and \(\{y_n\}\) do not.
Suppose that \(f\) is a function defined by :
\(f(x) = \frac{x}{3 - 2x}, \ 0 < x < 1\)
If \(g(x) = f''(x) + f'(x) - [f'(x)]^2\), then \(g(1)\) is equal to :
Consider the function \(f\) defined by :
\(f(x) = \begin{cases} \frac{\lambda \cos(x)}{\pi/2 - x}, & x \neq \pi/2 \\ 1, & x = \pi/2 \end{cases}\)
What should be the value of \(\lambda\) so that \(f\) would be continuous at \(x = \pi/2\) ?
Assuming \((X - I)^{-1}\) exists, what will be \(I + X + X^2 + \ldots + X^{n-1}\) ?
Important Questions from Mixed Topic (MH-SET)
.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
Every OS has a ................ or .......... which primarily resides in the main memory of the computer to perform some of the basic functions of the OS.
(i) Kernel
(ii) Interface
(iii) Nucleus
(iv) Compiler
Codes:
Which of the following are referred to as low level language?
(i) Machine language
(ii) Assembly language
(iii) High level language
(iv) Natural language
Codes:
Match the following:
List I List II (a) New York Public Library (i) Shelv AR (b) Miami University Augmented Research Group (ii) Smart library (c) Oula University Library (iii) Wolf Walk (d) North Carolina State University (iv) Find the future Codes:
Which part of the QR code represents the encoded data?
