Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Random Variables & Distribution Functions
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables with E(X1) = 0 and Var(X1) = 1. Which of the following statements are true?
1
\(\rm \lim_{n\rightarrow \infty}P\left(\frac{\sqrt n \Sigma_{i=1}^nX_i}{\Sigma_{i=1}^nX_i^2}\le 0\right)=\frac{1}{2}\)
2
\(\frac{ \Sigma_{i=1}^nX_i}{\Sigma_{i=1}^nX_i^2}\) converges in probability to 0 as n → ∞
3
\(\rm \frac{1}{n}\Sigma_{i=1}^nX_i^2\) converges in probability to 1 as n → ∞
4
\(\rm \lim_{n\rightarrow \infty}P\left(\frac{ \Sigma_{i=1}^nX_i}{\sqrt n}\le 0\right)=\frac{1}{2}\)