Let Xnn ge 1 be a sequence of i.i.d. random variables with the common probability density function f(x) = 1/pi (1 + x^2), ; -infty < x < infty. Define Yn = 1/2 + 1/pi tan^-1(Xn) for n = 1, 2, dots. Then which one of the following options is correct?

(A) 1/n sumi=1^n Yi xrightarrowP 1/2 as n to infty

Q16Section ST (Statistics (Q11–Q65))16 of 65

SLLN for Cauchy-transformed sample

Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables with the common probability density function $f(x) = \frac{1}{\pi (1 + x^2)}, \; -\infty < x < \infty$ . Define $Y_n = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(X_n)$ for $n = 1, 2, \dots$ . Then which one of the following options is correct?

Pick an option to check your answer. In a mock test a wrong answer here costs 1/3 mark.

Keyboard: A–D answer · S solution · ←/→ previous/next